# Using the Ratio Test to Determine if a Series Converges #3 (Factorials)

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

• ##### What is a Sequence? Basic Sequence Info
Patrick Jones

What 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.

• ##### Sequences - Examples showing convergence or divergence
Patrick Jones

Sequences - Examples showing convergence or divergence.

• ##### Summation Notation
Patrick Jones

Summation Notation - Summation Notation is explained using a bunch of examples!

• ##### What is a Series
Patrick Jones

What 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.

• ##### Geometric Series and the Test for Divergence - Part 1
Patrick Jones

Geometric Series and the Test for Divergence - This video has follow up examples!

• ##### Geometric Series and the Test for Divergence - Part 2
Patrick Jones

Geometric Series and the Test for Divergence - Part 2. In this video, I finish off the problem that got cut off!

• ##### Geometric Series - Expressing a Decimal as a Rational Number
Patrick Jones

Geometric 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.

• ##### Telescoping Series Example
Patrick Jones

Telescoping Series Example - Finding the sum of a telescoping series.

• ##### Showing a Series Diverges using Partial Sums
Patrick Jones

Showing a Series Diverges using Partial Sums

• ##### Using the Integral Test for Series
Patrick Jones

Using the Integral Test for Series - One complete example. I justify EVERYTHING!

• ##### Remainder Estimate for the Integral Test
Patrick Jones

Remainder 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.

• ##### Limit Comparison Test and Direct Comparison Test (Part 1)
Patrick Jones

Limit Comparison Test and Direct Comparison Test - Using the Limit Comparison and Direct Comparison Test to Determine if a Series Converges or Diverges.

• ##### Limit Comparison Test and Direct Comparison Test (Part 2)
Patrick Jones

Using the Limit Comparison Test to Determine if a series converges or diverges.

• ##### Alternating Series
Patrick Jones

Basic Idea of how to show an alternating series converges. A very EASY example is shown!

• ##### More Alternating Series Examples
Patrick Jones

More Alternating Series Examples - Finding whether a given alternating series converges or diverges.

• ##### Alternating Series Estimation Theorem
Patrick Jones

Alternating Series Estimation Theorem -The basic idea along with a couple of examples are shown.

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

Using the Ratio Test to Determine if a Series Converges or Diverges - Two examples are shown. In another video, two more examples are shown!

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

Using the Ratio Test to Determine if a Series Converges #2 - Two more examples are shown using the Ratio Test.

• ##### Using the Ratio Test to Determine if a Series Converges #3 (Factorials)
Patrick Jones

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

• ##### Root Test for Series
Patrick Jones

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

• ##### Strategy for Testing Series - Series Practice Problems
Patrick Jones

Strategy 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.

• ##### Absolute Convergence, Conditional Convergence and Divergence
Patrick Jones

Absolute Convergence, Conditional Convergence and Divergence for series. In this video, I give the basic result and do 3 examples!

• ##### Power Series Representation of Functions
Patrick Jones

Power 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

• ##### Power Series - Finding the Interval of Convergence
Patrick Jones

Power Series - Finding the Interval of Convergence - Two complete examples are shown!

• ##### Radius of Convergence for a Power Series
Patrick Jones

Radius 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.

• ##### Differentiating and Integrating Power Series
Patrick Jones

Differentiating 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.

• ##### Finding the Sum of a Series by Differentiating
Patrick Jones

Finding 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.

• ##### Finding Power Series by Differentiation - 3 examples
Patrick Jones

Finding 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.

• ##### Integrating a Power Series
Patrick Jones

In 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.

• ##### Integrating a Power Series, Example 2
Patrick Jones

Just another example of integrating a power series to find a new power series representation.

• ##### Finding Interval of Convergence for a Given Power Series Representation
Patrick Jones

Finding Interval of Convergence for a Given Power Series Representation. In this video, we find the interval of convergence for a given power series.

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

Interval 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.

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

Interval 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.

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

Interval and Radius of Convergence for a Series, Ex 5. Just another example of finding the radius and interval of convergence for a power series.

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

Interval and Radius of Convergence for a Series, Ex 6. Just another example of finding the radius and interval of convergence for a power series.

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

Interval and Radius of Convergence for a Series, Ex 7. Just another example of finding the radius and interval of convergence for a power series

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

Interval and Radius of Convergence for a Series, Ex 9. Just another example of finding the radius and interval of convergence for a power series.

• ##### Finding a New Power Series by Manipulating a Known Power Series
Patrick Jones

Finding 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.

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

1/(1-x) - Another Ex 1. Here I illustrate the idea of creating new power series by manipulating known power series.

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

Finding 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.

• ##### Finding a Maclaurin Series Expansion - Another Example 1
Patrick Jones

Finding a Maclaurin Series Expansion - Another Example 1. In this video, I find a Maclaurin series expansion for 1/(1-x).

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

Taylor's Remainder Theorem - Finding the Remainder, Ex 1. In this example, I use Taylor's Remainder Theorem to find an expression for the remainder.

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

Taylor's Remainder Theorem - Finding the Remainder, Ex 2. In this example, I use Taylor's Remainder Theorem to find an expression for the remainder.

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

Taylor's Remainder Theorem - Finding the Remainder, Ex 3. In this example, I use Taylor's Remainder Theorem to find an expression for the remainder.

• ##### Finding a Maclaurin Polynomial - Ex 1
Patrick Jones

Finding a Maclaurin Polynomial - Ex 1. In this example, I find a degree three Maclaurin polynomial to approximate e^(4x).

• ##### Finding a Maclaurin Polynomial - Ex 2
Patrick Jones

Finding a Maclaurin Polynomial - Ex 2. In this example, I find a degree three Maclaurin polynomial to approximate a function.

• ##### Finding a Taylor Polynomial to Approximate a Function, Ex 1
Patrick Jones

In this video, I find a degree 3 Taylor Polynomial to approximate sin(x).

• ##### The Root Test - Another Example, #3
Patrick Jones

The Root Test - Another Example, #3. Just another example showing that a series converges or diverges using the root test.

• ##### Finding a Taylor Polynomial to Approximate a Function, Ex 2
Patrick Jones

Finding a Taylor Polynomial to Approximate a Function, Ex 2. In this video, I find a degree 4 Taylor Polynomial to approximate a function.

• ##### Finding a Taylor Polynomial to Approximate a Function, Ex 3
Patrick Jones

Finding a Taylor Polynomial to Approximate a Function, Ex 3. In this example, I find a degree three Taylor Polynomial to approximate a given function.

• ##### Finding a Taylor Polynomial to Approximate a Function, Ex 4
Patrick Jones

Finding a Taylor Polynomial to Approximate a Function, Ex 4. In this example, I find a degree 2 Taylor polynomial to approximate cot(x).

• ##### The Root Test - Another Example, #2
Patrick Jones

The Root Test - Another Example, #2. Just another example showing that a series converges or diverges using the root test.

• ##### The Ratio Test , Another Example #1
Patrick Jones

The Ratio Test , Another Example #1. Just another example of using the ratio test to determine if a series converges or diverges.

• ##### The Ratio Test , Another Example #2
Patrick Jones

The Ratio Test , Another Example #2. Just another example of using the ratio test to determine if a series converges or diverges.

• ##### The Ratio Test , Another Example #3
Patrick Jones

The Ratio Test , Another Example #3. Just another example showing that series converges or diverges using the ratio test.

• ##### The Ratio Test , Another Example #4
Patrick Jones

The Ratio Test , Another Example #4. Just another example showing that series converges or diverges using the ratio test.

• ##### Absolute Convergence, Conditional Convergence, Another Example 1
Patrick Jones

Absolute 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.

• ##### Absolute Convergence, Conditional Convergence, Another Example 2
Patrick Jones

Absolute 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.

• ##### Absolute Convergence, Conditional Convergence, Another Example 3
Patrick Jones

Absolute 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.

• ##### Alternating Series - Another Example 1
Patrick Jones

Alternating Series - Another Example 1. In this video, I show that an alternating series converges or diverges.

• ##### Alternating Series - Another Example 2
Patrick Jones

Alternating Series - Another Example 2. In this video, I show that an alternating series converges or diverges.

• ##### Alternating Series - Another Example 3
Patrick Jones

Alternating Series - Another Example 3. Just another example showing an alternating series converges or diverges.

• ##### Alternating Series - Another Example 4
Patrick Jones

Alternating Series - Another Example 4. Just another example showing an alternating series converges or diverges.

• ##### Intro to Summation Notation and Infinite Series, Ex 1
Patrick Jones

Intro 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.

• ##### Limit Comparison Test for Series - Another Example 1
Patrick Jones

Limit 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.

• ##### Limit Comparison Test for Series - Another Example 2
Patrick Jones

Limit 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.

• ##### Limit Comparison Test for Series - Another Example 3
Patrick Jones

Limit 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.

• ##### Limit Comparison Test for Series - Another Example 4
Patrick Jones

Limit 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.

• ##### Limit Comparison Test for Series - Another Example 5
Patrick Jones

Limit 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.

• ##### Intro to Monotonic and Bounded Sequences, Ex 1
Patrick Jones

Intro to Monotonic and Bounded Sequences, Ex 1. Just a little question to decide if a sequence is monotonic and/or bounded.

• ##### The Squeeze Theorem and Absolute Value Theorem, #1
Patrick Jones

The 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.

• ##### The Squeeze Theorem and Absolute Value Theorem, #2
Patrick Jones

The 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.

• ##### The Squeeze Theorem and Absolute Value Theorem, #3
Patrick Jones

The Squeeze Theorem and Absolute Value Theorem, #3. Here we find the limit of another sequence by using the squeeze theorem.

• ##### Finding the Limit of a Sequence, 3 more examples
Patrick Jones

Finding 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.

• ##### Multiplication and Division of Power Series
Patrick Jones

Multiplication and Division of Power Series - Using Multiplication and Division of Power Series to derive Power Series Representations for other functions!

• ##### Taylor and Maclaurin Series - Example 1
Patrick Jones

Taylor 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!

• ##### Taylor / Maclaurin Series for Sin (x)
Patrick Jones

Taylor / 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.

• ##### Taylor and Maclaurin Series - Example 2
Patrick Jones

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

• ##### Using Series to Evaluate Limits
Patrick Jones

Using 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.

• ##### Using Maclaurin/Taylor Series to Approximate a Definite Integral
Patrick Jones

Using 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.

• ##### The Binomial Series - Example 1
Patrick Jones

Using the Binomial Series to derive power series representations for another function. One example is shown!

• ##### The Binomial Series - Example 2
Patrick Jones

Using 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).

• ##### Integrating a Function as a Power Series
Patrick Jones

Here we integrate: [ e^(x^2) ] / x by integrating the corresponding power series representation for the function.

• ##### Finding a Power Series Representation for a Logarithm Function
Patrick Jones

Finding a Power Series Representation for a Logarithm. Here we find a power series representation for a function involving the natural logarithm function.

• ##### Finding a Function to Match a Given Power Series by Integrating
Patrick Jones

In 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.

• ##### Finding a Power Series by Differentiation
Patrick Jones

In 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)

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

Interval 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.

• ##### Direct Comparison Test - Another Example 2
Patrick Jones

Direct Comparison Test - Another Example 2. In this video I show that another series converges or diverges using the direct comparison theorem.

• ##### Direct Comparison Test - Another Example 1
Patrick Jones

Direct Comparison Test - Another Example 1. In this video I show that another series converges or diverges using the direct comparison theorem.

• ##### Direct Comparison Test - Another Example 3
Patrick Jones

Direct Comparison Test - Another Example 3. In this video I show that another series converges or diverges using the direct comparison theorem.

• ##### P-Series
Patrick Jones

In 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.

• ##### Integral Test to Evaluate Series, Ex 4
Patrick Jones

Integral Test to Evaluate Series, Ex 4. One more example using the integral test to evaluate series.

• ##### Integral Test to Evaluate Series, Ex 3
Patrick Jones

Integral Test to Evaluate Series, Ex 3. In this video, I show that another series converges or diverges by using the integral test for series.

• ##### Integral Test to Evaluate Series, Ex 2
Patrick Jones

Integral 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).

• ##### Integral Test to Evaluate Series, Ex 1
Patrick Jones

Integral Test to Evaluate Series, Ex 1. In this video, I show how to evaluate a series by using the integral test.

• ##### Telescoping Series ,Showing Divergence Using Partial Sums
Patrick Jones

Telescoping 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.

• ##### Telescoping Series , Finding the Sum, Example 1
Patrick Jones

Telescoping 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.

• ##### Sum of an Infinite Geometric Series, Ex 3
Patrick Jones

Sum 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.

• ##### Sum of an Infinite Geometric Series, Ex 2
Patrick Jones

Sum 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.

• ##### Sum of an Infinite Geometric Series, Ex 1
Patrick Jones

Sum of an Infinite Geometric Series, Ex 1. In this video, I show how to find the sum of a convergent infinite series.

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

Writing 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.

• ##### Finding a Formula for a Partial Sum of a Telescoping Series
Patrick Jones

Finding 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)

• ##### Writing a Geometric Series using Sigma / Summation Notation
Patrick Jones

Writing 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.

• ##### Test for Divergence for Series, Two Examples
Patrick Jones

Test 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.

• ##### Direct Comparison Test - Another Example 4
Patrick Jones

Direct Comparison Test - Another Example 4. In this video I show that another series converges or diverges using the direct comparison theorem.

• ##### Direct Comparison Test - Another Example 5
Patrick Jones

Direct Comparison Test - Another Example 5. In this video I show that another series converges or diverges using the direct comparison theorem.

• ##### The Root Test - Another Example, #1
Patrick Jones

The Root Test - Another Example, #1. Just another example showing that a series converges or diverges using the root test.

• ##### Alternating Series - Error Estimation #2
Patrick Jones

Alternating 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.

• ##### Alternating Series - Error Estimation
Patrick Jones

Alternating 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.

• ##### Limit Comparison Test for Series - Another Example 6
Patrick Jones

Limit 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.

• ##### Limit Comparison Test for Series - Another Example 7
Patrick Jones

Limit 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.

• ##### Limit Comparison Test for Series - Another Example 8
Patrick Jones

Limit 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.
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