Calculus 2 Lecture 9.7: Power Series, Calculus of Power Series, Ratio Test for Int. of Convergence

Calculus 2 Lecture 9.7: Power Series, Calculus of Power Series, Using Ratio Test to Find Interval of Convergence

[RequiemForABuckeye]

quick head nod if you're ok with this video

[SI]

0:00:00 Introduction and definition of Power Series. Idea of center point.
0:07:05 Example (-1^n)(x^n)/(n!) - PS centered at 0
0:10:10 Example (-1^n)(x - pi/4)^(2n+1)/(2n+1)! - PS centered at pi/4
0:14:00 The idea that these PS are FUNCTIONS of x. I.e., the SUM of the PS is the OUTPUT of the FUNCTION. But only if/where the PS is convergent.
0:21:35 Example x^n - Geometric Series - definition of common ratio
0:28:04 Example x^n - The form of the function represented by this PS
0:33:30 More about the center point, the domain, the interval of convergence
0:36:24 The 3 possible conditions in which PS may converge (Type "a", "b", "c"). Radius of Convergence definition. Interval of Convergence definition.
0:44:14 Example (n!)(x^n) - Why we will almost always use the Ratio Test for IOC. (Especially when factorials involved)
0:48:05 Example (n!)(x^n) - Why we can move everything other than the "x" outside the absolute value symbol
0:55:18 Example (n!)(x^n) - Why this example has no endpoints and no ROC or IOC, just a single convergent value. (Type "a")
0:56:37 Example (-1^n)(x^2n)/(2n)! - Ratio Test
1:06:09 Example (-1^n)(x^2n)/(2n)! - Why we don't need the absolute value sign after simplifying
1:08:06 Example (-1^n)(x^2n)/(2n)! - This example converges for all x (Type "b")
1:11:55 Example (x^n)/n - Ratio Test. Why the other tests are not optimal. This is the first example where we must test the endpoints (Type "c").
1:17:55 Example (x^n)/n - More insight on why we use the Ratio Test.
1:20:10 Example (x^n)/n - How we determine the ROC and the endpoints. Then, checking the endpoints tells us the IOC.
1:22:49 Example (x^n)/n - Endpoint 1 (Alternating Series Test)
1:24:55 Example (x^n)/n - Endpoint 2 (Harmonic P-Series)
1:28:00 Example (x-2)^n / (n^2 * 3^2) - Ratio Test
1:35:20 Example (x-2)^n / (n^2 * 3^2) - R.O.C. when not centered at 0, and finding endpoints of such interval
1:37:30 Example (x-2)^n / (n^2 * 3^2) - Endpoints (Alternating Series test, P-series)
1:41:18 Example (-1^n)(2^n)(x^n)/sqrt(n+1) - Ratio Test
1:51:10 Example (-1^n)(2^n)(x^n)/sqrt(n+1) - Endpoint 1 (Limit Comparison Test)
1:56:10 Example (-1^n)(2^n)(x^n)/sqrt(n+1) - Manipulating the starting n value together with the series form to make a certain test valid
1:58:40 Example (-1^n)(2^n)(x^n)/sqrt(n+1) - Endpoint 2 (Alternating Series Test)
2:03:12 Derivatives & Integrals of Power Series introduction. Basically we have a series (which IS a function) - we expand it out - we take the derivative - then we collapse it back in to a new series formula.
2:05:52 Derivatives
2:12:35 Integrals
2:15:55 Losing convergent endpoints via derivatives, gaining convergent endpoints via integrals
2:17:04 Example - Representing ln(1-x) on (-1, 1) as a Power Series (its derivative has form of Geometric Series). Discussion of why the interval range affects the validity of this method.

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