Here we integrate: [ e^(x^2) ] / x by integrating the corresponding power series representation for the function.
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.
Summation Notation - Summation Notation is explained using a bunch of examples!
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 - This video has follow up examples!
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. In this video I show how to convert the number 5.1212121212..... into a fraction using geometric series.
Telescoping Series Example - Finding the sum of a telescoping series.
Showing a Series Diverges using Partial Sums
Using the Integral Test for Series - One complete example. I justify EVERYTHING!
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 - Using the Limit Comparison and Direct Comparison Test to Determine if a Series Converges or Diverges.
Using the Limit Comparison Test to Determine if a series converges or diverges.
Basic Idea of how to show an alternating series converges. A very EASY example is shown!
More Alternating Series Examples - Finding whether a given alternating series converges or diverges.
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 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 - Two more examples are shown using the Ratio Test.
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 - 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. 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 for series. In this video, I give the basic result and do 3 examples!
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 - Two complete examples are shown!
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 - 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. 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. In this video, I take the derivative of the power series for 1/(1-x) to create new power series representations.
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.
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. In this video, we find the interval of convergence for a given power series.
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. 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. 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. 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. 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. 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. In this video, we are given the power series for e^x and use that to find a new power series.
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. 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. In this video, I find a Maclaurin series expansion for 1/(1-x).
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. 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. In this example, I use Taylor's Remainder Theorem to find an expression for the remainder.
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. In this example, I find a degree three Maclaurin polynomial to approximate a function.
In this video, I find a degree 3 Taylor Polynomial to approximate sin(x).
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. In this video, I find a degree 4 Taylor Polynomial to approximate a function.
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. In this example, I find a degree 2 Taylor polynomial to approximate cot(x).
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. Just another example of using the ratio test to determine if a series converges or diverges.
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. Just another example showing that series converges or diverges using the ratio test.
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. 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. 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. Here we looks at some more examples to determine whether a series is absolutely convergent, conditionally convergent or divergent.
Alternating Series - Another Example 1. In this video, I show that an alternating series converges or diverges.
Alternating Series - Another Example 2. In this video, I show that an alternating series converges or diverges.
Alternating Series - Another Example 3. Just another example showing an alternating series converges or diverges.
Alternating Series - Another Example 4. Just another example showing an alternating series converges or diverges.
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. 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. 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. 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. 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. 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. Just a little question to decide if a sequence is monotonic and/or bounded.
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. 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. Here we find the limit of another sequence by using the squeeze theorem.
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 - Using Multiplication and Division of Power Series to derive Power Series Representations for other functions!
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). 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 - A complete example of finding a Taylor series for the function ln(x) centered at a = 2 is shown.
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 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.
Using the Binomial Series to derive power series representations for another function. One example is shown!
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).
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. Here we find a power series representation for a function involving the natural logarithm function.
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.
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. In this video, I show another example of finding the interval and radius of convergence for a series.
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. In this video I show that another series converges or diverges using the direct comparison theorem.
Direct Comparison Test - Another Example 3. In this video I show that another series converges or diverges using the direct comparison theorem.
Integral Test to Evaluate Series, Ex 4. One more example using the integral test to evaluate series.
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. 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. In this video, I show how to evaluate a series by using the integral test.
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. 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. 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. 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. 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. 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. 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. 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. 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. In this video I show that another series converges or diverges using the direct comparison theorem.
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. Just another example showing that a series converges or diverges using the root test.
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. 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. 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. 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. In this video, I use the limit comparison test to determine whether or not a given series converges or diverges.