Using the Ratio Test to Determine if a Series Converges #3 - Another example using the ratio test of a series that has factorials.

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### What is a Sequence? Basic Sequence Info

00:11:26Patrick JonesWhat is a Sequence? Basic Sequence Info. In this video, I discuss what a sequence is, what it means for a sequence to converge or diverge, and do some examples.

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### Sequences - Examples showing convergence or divergence

00:06:12Patrick JonesSequences - Examples showing convergence or divergence.

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### Summation Notation

00:10:16Patrick JonesSummation Notation - Summation Notation is explained using a bunch of examples!

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### What is a Series

00:12:24Patrick JonesWhat is a Series - I try to give a basic idea of what a series is. I also briefly discuss geometric series and the test for divergence.

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### Geometric Series and the Test for Divergence - Part 1

00:09:57Patrick JonesGeometric Series and the Test for Divergence - This video has follow up examples!

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### Geometric Series and the Test for Divergence - Part 2

00:02:55Patrick JonesGeometric Series and the Test for Divergence - Part 2. In this video, I finish off the problem that got cut off!

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### Geometric Series - Expressing a Decimal as a Rational Number

00:07:17Patrick JonesGeometric Series - Expressing a Decimal as a Rational Number. In this video I show how to convert the number 5.1212121212..... into a fraction using geometric series.

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### Telescoping Series Example

00:04:39Patrick JonesTelescoping Series Example - Finding the sum of a telescoping series.

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### Showing a Series Diverges using Partial Sums

00:07:36Patrick JonesShowing a Series Diverges using Partial Sums

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### Using the Integral Test for Series

00:08:20Patrick JonesUsing the Integral Test for Series - One complete example. I justify EVERYTHING!

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### Remainder Estimate for the Integral Test

00:07:46Patrick JonesRemainder Estimate for the Integral Test. In this video, I show the Remainder Estimate for the Integral Test, show a series converges using the integral test, and then estimate the error of that series using the first 10 terms.

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### Limit Comparison Test and Direct Comparison Test (Part 1)

00:08:53Patrick JonesLimit Comparison Test and Direct Comparison Test - Using the Limit Comparison and Direct Comparison Test to Determine if a Series Converges or Diverges.

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### Limit Comparison Test and Direct Comparison Test (Part 2)

00:07:35Patrick JonesUsing the Limit Comparison Test to Determine if a series converges or diverges.

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### Alternating Series

00:06:01Patrick JonesBasic Idea of how to show an alternating series converges. A very EASY example is shown!

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### More Alternating Series Examples

00:07:18Patrick JonesMore Alternating Series Examples - Finding whether a given alternating series converges or diverges.

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### Alternating Series Estimation Theorem

00:09:47Patrick JonesAlternating Series Estimation Theorem -The basic idea along with a couple of examples are shown.

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### Using the Ratio Test to Determine if a Series Converges #1

00:07:38Patrick JonesUsing the Ratio Test to Determine if a Series Converges or Diverges - Two examples are shown. In another video, two more examples are shown!

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### Using the Ratio Test to Determine if a Series Converges #2

00:09:24Patrick JonesUsing the Ratio Test to Determine if a Series Converges #2 - Two more examples are shown using the Ratio Test.

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### Root Test for Series

00:10:07Patrick JonesRoot Test for Series - Using the Root Test to Determine if a Series Converges or Diverges! The test along with 3 full examples are shown!

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### Strategy for Testing Series - Series Practice Problems

00:12:47Patrick JonesStrategy for Testing Series - Series Practice Problems. In this video, I run through 14 series problems, discussing what I would do to show they converge or diverge. I try to show what I think about when I look at a series, and what 'sticks out to me' as to how I should justify things.

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### Absolute Convergence, Conditional Convergence and Divergence

00:11:21Patrick JonesAbsolute Convergence, Conditional Convergence and Divergence for series. In this video, I give the basic result and do 3 examples!

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### Power Series Representation of Functions

00:10:09Patrick JonesPower Series Representation of Functions - In this video I manipulate the power series representation of 1/(1-x) to derive power series representations of other functions

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### Power Series - Finding the Interval of Convergence

00:09:47Patrick JonesPower Series - Finding the Interval of Convergence - Two complete examples are shown!

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### Radius of Convergence for a Power Series

00:01:48Patrick JonesRadius of Convergence for a Power Series - In this video, I discuss how to find the radius of converge. It is easy after you find the INTERVAL of convergence.

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### Differentiating and Integrating Power Series

00:10:10Patrick JonesDifferentiating and Integrating Power Series - Two examples are shown of integrating or differentiating a known power series to derive a power series representation for a new function.

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### Finding the Sum of a Series by Differentiating

00:04:01Patrick JonesFinding the Sum of a Series by Differentiating. Ok, I think this is sort of a 'tricky' problem! Here we find the sum of a series by differentiating a known power series to get to original series into a more recognizable form.

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### Finding Power Series by Differentiation - 3 examples

00:12:13Patrick JonesFinding Power Series by Differentiation - 3 examples. In this video, I take the derivative of the power series for 1/(1-x) to create new power series representations.

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### Integrating a Power Series

00:03:04Patrick JonesIn this video, I do a simple example of integrating a power series. This is a useful trick as we often do this to find new power series representations by integrating known power series representations.

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### Integrating a Power Series, Example 2

00:03:13Patrick JonesJust another example of integrating a power series to find a new power series representation.

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### Finding Interval of Convergence for a Given Power Series Representation

00:03:14Patrick JonesFinding Interval of Convergence for a Given Power Series Representation. In this video, we find the interval of convergence for a given power series.

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### Interval and Radius of Convergence for a Series, Ex 3

00:02:15Patrick JonesInterval and Radius of Convergence for a Series, Ex 3. In this video, I show another example of finding the interval and radius of convergence for a series.

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### Interval and Radius of Convergence for a Series, Ex 4

00:05:21Patrick JonesInterval and Radius of Convergence for a Series, Ex 4. Just another example where I find the radius and interval of convergence for a power series.

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### Interval and Radius of Convergence for a Series, Ex 5

00:05:21Patrick JonesInterval and Radius of Convergence for a Series, Ex 5. Just another example of finding the radius and interval of convergence for a power series.

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### Interval and Radius of Convergence for a Series, Ex 6

00:06:19Patrick JonesInterval and Radius of Convergence for a Series, Ex 6. Just another example of finding the radius and interval of convergence for a power series.

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### Interval and Radius of Convergence for a Series, Ex 7

00:05:12Patrick JonesInterval and Radius of Convergence for a Series, Ex 7. Just another example of finding the radius and interval of convergence for a power series

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### Interval and Radius of Convergence for a Series, Ex 9

00:08:21Patrick JonesInterval and Radius of Convergence for a Series, Ex 9. Just another example of finding the radius and interval of convergence for a power series.

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### Finding a New Power Series by Manipulating a Known Power Series

00:02:12Patrick JonesFinding a New Power Series by Manipulating a Known Power Series. In this video, we are given the power series for e^x and use that to find a new power series.

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### Finding Power Series Representations by Manipulating 1/(1-x) - Another Ex 1

00:01:31Patrick Jones1/(1-x) - Another Ex 1. Here I illustrate the idea of creating new power series by manipulating known power series.

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### Finding a New Power Series by Manipulating a Known Power Series, Ex 2

00:07:07Patrick JonesFinding a New Power Series by Manipulating a Known Power Series. In this video, we are given the power series for cos(x) and use that to find a new power series representation for (cos x) ^ 2.

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### Finding a Maclaurin Series Expansion - Another Example 1

00:04:50Patrick JonesFinding a Maclaurin Series Expansion - Another Example 1. In this video, I find a Maclaurin series expansion for 1/(1-x).

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### Taylor's Remainder Theorem - Finding the Remainder, Ex 1

00:02:22Patrick JonesTaylor's Remainder Theorem - Finding the Remainder, Ex 1. In this example, I use Taylor's Remainder Theorem to find an expression for the remainder.

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### Taylor's Remainder Theorem - Finding the Remainder, Ex 2

00:03:44Patrick JonesTaylor's Remainder Theorem - Finding the Remainder, Ex 2. In this example, I use Taylor's Remainder Theorem to find an expression for the remainder.

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### Taylor's Remainder Theorem - Finding the Remainder, Ex 3

00:04:37Patrick JonesTaylor's Remainder Theorem - Finding the Remainder, Ex 3. In this example, I use Taylor's Remainder Theorem to find an expression for the remainder.

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### Finding a Maclaurin Polynomial - Ex 1

00:03:04Patrick JonesFinding a Maclaurin Polynomial - Ex 1. In this example, I find a degree three Maclaurin polynomial to approximate e^(4x).

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### Finding a Maclaurin Polynomial - Ex 2

00:04:03Patrick JonesFinding a Maclaurin Polynomial - Ex 2. In this example, I find a degree three Maclaurin polynomial to approximate a function.

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### Finding a Taylor Polynomial to Approximate a Function, Ex 1

00:05:26Patrick JonesIn this video, I find a degree 3 Taylor Polynomial to approximate sin(x).

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### The Root Test - Another Example, #3

00:01:42Patrick JonesThe Root Test - Another Example, #3. Just another example showing that a series converges or diverges using the root test.

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### Finding a Taylor Polynomial to Approximate a Function, Ex 2

00:02:58Patrick JonesFinding a Taylor Polynomial to Approximate a Function, Ex 2. In this video, I find a degree 4 Taylor Polynomial to approximate a function.

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### Finding a Taylor Polynomial to Approximate a Function, Ex 3

00:05:08Patrick JonesFinding a Taylor Polynomial to Approximate a Function, Ex 3. In this example, I find a degree three Taylor Polynomial to approximate a given function.

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### Finding a Taylor Polynomial to Approximate a Function, Ex 4

00:05:33Patrick JonesFinding a Taylor Polynomial to Approximate a Function, Ex 4. In this example, I find a degree 2 Taylor polynomial to approximate cot(x).

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### The Root Test - Another Example, #2

00:03:46Patrick JonesThe Root Test - Another Example, #2. Just another example showing that a series converges or diverges using the root test.

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### The Ratio Test , Another Example #1

00:03:50Patrick JonesThe Ratio Test , Another Example #1. Just another example of using the ratio test to determine if a series converges or diverges.

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### The Ratio Test , Another Example #2

00:02:01Patrick JonesThe Ratio Test , Another Example #2. Just another example of using the ratio test to determine if a series converges or diverges.

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### The Ratio Test , Another Example #3

00:03:47Patrick JonesThe Ratio Test , Another Example #3. Just another example showing that series converges or diverges using the ratio test.

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### The Ratio Test , Another Example #4

00:03:04Patrick JonesThe Ratio Test , Another Example #4. Just another example showing that series converges or diverges using the ratio test.

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### Absolute Convergence, Conditional Convergence, Another Example 1

00:02:35Patrick JonesAbsolute Convergence, Conditional Convergence, Another Example 1. Here we looks at some more examples to determine whether a series is absolutely convergent, conditionally convergent or divergent.

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### Absolute Convergence, Conditional Convergence, Another Example 2

00:02:38Patrick JonesAbsolute Convergence, Conditional Convergence, Another Example 2. Here we looks at some more examples to determine whether a series is absolutely convergent, conditionally convergent or divergent.

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### Absolute Convergence, Conditional Convergence, Another Example 3

00:02:09Patrick JonesAbsolute Convergence, Conditional Convergence, Another Example 3. Here we looks at some more examples to determine whether a series is absolutely convergent, conditionally convergent or divergent.

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### Alternating Series - Another Example 1

00:02:00Patrick JonesAlternating Series - Another Example 1. In this video, I show that an alternating series converges or diverges.

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### Alternating Series - Another Example 2

00:01:29Patrick JonesAlternating Series - Another Example 2. In this video, I show that an alternating series converges or diverges.

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### Alternating Series - Another Example 3

00:02:01Patrick JonesAlternating Series - Another Example 3. Just another example showing an alternating series converges or diverges.

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### Alternating Series - Another Example 4

00:02:35Patrick JonesAlternating Series - Another Example 4. Just another example showing an alternating series converges or diverges.

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### Intro to Summation Notation and Infinite Series, Ex 1

00:01:38Patrick JonesIntro to Summation Notation and Infinite Series, Ex 1. Forgot about summation notation? Here is a little refresher example! Notice the ' i ' has NOTHING to do with complex numbers.

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### Limit Comparison Test for Series - Another Example 1

00:03:03Patrick JonesLimit Comparison Test for Series - Another Example 1. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges.

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### Limit Comparison Test for Series - Another Example 2

00:02:06Patrick JonesLimit Comparison Test for Series - Another Example 2. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges.

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### Limit Comparison Test for Series - Another Example 3

00:04:01Patrick JonesLimit Comparison Test for Series - Another Example 3. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges.

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### Limit Comparison Test for Series - Another Example 4

00:01:54Patrick JonesLimit Comparison Test for Series - Another Example 4. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges.

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### Limit Comparison Test for Series - Another Example 5

00:02:53Patrick JonesLimit Comparison Test for Series - Another Example 5. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges.

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### Intro to Monotonic and Bounded Sequences, Ex 1

00:03:42Patrick JonesIntro to Monotonic and Bounded Sequences, Ex 1. Just a little question to decide if a sequence is monotonic and/or bounded.

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### The Squeeze Theorem and Absolute Value Theorem, #1

00:03:04Patrick JonesThe Squeeze Theorem and Absolute Value Theorem, #1. Here we look at finding the limits of some sequences by using the squeeze and / or absolute value theorems.

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### The Squeeze Theorem and Absolute Value Theorem, #2

00:03:33Patrick JonesThe Squeeze Theorem and Absolute Value Theorem, #2. Just another example of finding the limit of a sequence by using the squeeze and / or absolute value theorem.

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### The Squeeze Theorem and Absolute Value Theorem, #3

00:02:36Patrick JonesThe Squeeze Theorem and Absolute Value Theorem, #3. Here we find the limit of another sequence by using the squeeze theorem.

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### Finding the Limit of a Sequence, 3 more examples

00:03:24Patrick JonesFinding the Limit of a Sequence, 3 more examples, #1. Just another example of finding the limit of a sequence by taking a limit as n approaches infinity.

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### Multiplication and Division of Power Series

00:08:56Patrick JonesMultiplication and Division of Power Series - Using Multiplication and Division of Power Series to derive Power Series Representations for other functions!

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### Taylor and Maclaurin Series - Example 1

00:06:30Patrick JonesTaylor and Maclaurin Series - An example of finding the Maclaurin series for a function is shown. In another video, I will find a Taylor series expansion, so look for that one too!

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### Taylor / Maclaurin Series for Sin (x)

00:05:50Patrick JonesTaylor / Maclaurin Series for Sin (x). In this video, I show how to find the power series representation for sin(x) using a Taylor/Maclaurin series expansion.

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### Taylor and Maclaurin Series - Example 2

00:09:45Patrick JonesTaylor and Maclaurin Series - A complete example of finding a Taylor series for the function ln(x) centered at a = 2 is shown.

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### Using Series to Evaluate Limits

00:04:07Patrick JonesUsing Series to Evaluate Limits - In this example I show how one can use a series expansion and a bit of algebra to calculate a limit.

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### Using Maclaurin/Taylor Series to Approximate a Definite Integral

00:10:44Patrick JonesUsing Maclaurin/Taylor Series to Approximate a Definite Integral to a Desired Accuracy. In this video, I use Maclaurin/Taylor series and the Alternating Series Estimation Theorem to approximate a definite integral to within a desired accuracy.

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### The Binomial Series - Example 1

00:09:10Patrick JonesUsing the Binomial Series to derive power series representations for another function. One example is shown!

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### The Binomial Series - Example 2

00:09:13Patrick JonesUsing the Binomial Series to derive power series representations for another function. I show how to use Binomial Series to find a Maclaurin series representation for arcsin(x).

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### Integrating a Function as a Power Series

00:04:20Patrick JonesHere we integrate: [ e^(x^2) ] / x by integrating the corresponding power series representation for the function.

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### Finding a Power Series Representation for a Logarithm Function

00:09:40Patrick JonesFinding a Power Series Representation for a Logarithm. Here we find a power series representation for a function involving the natural logarithm function.

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### Finding a Function to Match a Given Power Series by Integrating

00:03:37Patrick JonesIn this example, we are given the power series representation for cos(x) and use that to find a function that corresponds to a new given series expansion.

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### Finding a Power Series by Differentiation

00:04:51Patrick JonesIn this video, we are given the power series representation for sin(x) and use that to find a power series representation for: xcos(x) – sin(x)

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### Interval and Radius of Convergence for a Series, Ex 2

00:05:26Patrick JonesInterval and Radius of Convergence for a Series, Ex 2. In this video, I show another example of finding the interval and radius of convergence for a series.

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### Direct Comparison Test - Another Example 2

00:03:08Patrick JonesDirect Comparison Test - Another Example 2. In this video I show that another series converges or diverges using the direct comparison theorem.

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### Direct Comparison Test - Another Example 1

00:02:34Patrick JonesDirect Comparison Test - Another Example 1. In this video I show that another series converges or diverges using the direct comparison theorem.

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### Direct Comparison Test - Another Example 3

00:02:29Patrick JonesDirect Comparison Test - Another Example 3. In this video I show that another series converges or diverges using the direct comparison theorem.

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### P-Series

00:02:54Patrick JonesIn this video, I look at a few VERY simple p-series. I give the formula that shows which p-series converge, and which diverge as well. NOTHING HEAVY!!! You can justify these results by using the integral test.

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### Integral Test to Evaluate Series, Ex 4

00:04:09Patrick JonesIntegral Test to Evaluate Series, Ex 4. One more example using the integral test to evaluate series.

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### Integral Test to Evaluate Series, Ex 3

00:04:53Patrick JonesIntegral Test to Evaluate Series, Ex 3. In this video, I show that another series converges or diverges by using the integral test for series.

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### Integral Test to Evaluate Series, Ex 2

00:05:37Patrick JonesIntegral Test to Evaluate Series, Ex 2. In this video, I show another example using the integral test to show that a series converges or diverges. I justify all the steps (continuous, positive, decreasing).

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### Integral Test to Evaluate Series, Ex 1

00:03:56Patrick JonesIntegral Test to Evaluate Series, Ex 1. In this video, I show how to evaluate a series by using the integral test.

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### Telescoping Series ,Showing Divergence Using Partial Sums

00:04:47Patrick JonesTelescoping Series ,Showing Divergence Using Partial Sums. Here I find a formula for a partial sum of a geometric series and show that the series diverges.

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### Telescoping Series , Finding the Sum, Example 1

00:07:00Patrick JonesTelescoping Series , Finding the Sum, Example 1. Here I find a formula for a series that is telescoping, use partial fractions to decompose the formula, look at partial sums, and take a limit to find the sum.

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### Sum of an Infinite Geometric Series, Ex 3

00:03:40Patrick JonesSum of an Infinite Geometric Series, Ex 3. In this video, I do yet one more example of finding the sum of a convergent infinite series.

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### Sum of an Infinite Geometric Series, Ex 2

00:03:12Patrick JonesSum of an Infinite Geometric Series, Ex 2. In this video, I show how to find the value of the sum of two convergent infinite series.

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### Sum of an Infinite Geometric Series, Ex 1

00:01:37Patrick JonesSum of an Infinite Geometric Series, Ex 1. In this video, I show how to find the sum of a convergent infinite series.

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### Writing a Geometric Series using Sigma / Summation Notation, Ex 2

00:03:07Patrick JonesWriting a Geometric Series using Sigma / Summation Notation, Ex 2. This video shows how to write an Infinite geometric series .using sigma / summation notation. I do not find the actual sum for this particular convergent geometric series.

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### Finding a Formula for a Partial Sum of a Telescoping Series

00:04:16Patrick JonesFinding a Formula for a Partial Sum of a Telescoping Series. To evaluate a telescoping series, one typically finds an expression for a partial sum, and then takes the limit of this partial sum. In this video, I show how I go about finding a formula for the partial sum. I do not take the limit to get a value of the corresponding infinite series (although you can do that by simply taking a limit at infinity of the final expression that I find)

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### Writing a Geometric Series using Sigma / Summation Notation

00:02:12Patrick JonesWriting a Geometric Series using Sigma / Summation Notation. This video shows how to write the infinite geometric series : 1 + 0.1 + 0.01 + 0.001+ ....using sigma / summation notation. I do not find the actual sum for this particular convergent geometric series.

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### Test for Divergence for Series, Two Examples

00:03:37Patrick JonesTest for Divergence for Series, Two Examples. In this video, I discuss the test for divergence and show two examples of series who diverge by using the test for divergence.

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### Direct Comparison Test - Another Example 4

00:03:39Patrick JonesDirect Comparison Test - Another Example 4. In this video I show that another series converges or diverges using the direct comparison theorem.

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### Direct Comparison Test - Another Example 5

00:03:31Patrick JonesDirect Comparison Test - Another Example 5. In this video I show that another series converges or diverges using the direct comparison theorem.

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### The Root Test - Another Example, #1

00:02:01Patrick JonesThe Root Test - Another Example, #1. Just another example showing that a series converges or diverges using the root test.

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### Alternating Series - Error Estimation #2

00:01:16Patrick JonesAlternating Series - Error Estimation #2. In this video, I sum up the first few terms of an alternating series and then I find the maximum error involved.

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### Alternating Series - Error Estimation

00:02:26Patrick JonesAlternating Series - Error Estimation. In this example, I find the number of terms required so that we can estimate the value of our convergent alternating series correct to two decimal places.

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### Limit Comparison Test for Series - Another Example 6

00:01:26Patrick JonesLimit Comparison Test for Series - Another Example 6. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges.

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### Limit Comparison Test for Series - Another Example 7

00:04:54Patrick JonesLimit Comparison Test for Series - Another Example 7. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges.

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### Limit Comparison Test for Series - Another Example 8

00:02:33Patrick JonesLimit Comparison Test for Series - Another Example 8. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges.