Calculus II: Differentiation and Integration of a power series

In this video, we discuss integration and differentiation of power series expressions and cover several examples on how to manipulate power series this way.

00:00 - Introduction
00:32 - Differentiating power series
04:52 - Integrating power series
07:28 - Example 1
09:13 - Example 2
15:00 - Example 3
19:26 - Example 4
31:31 - Example 5
42:15 - Example 6
45:41 - Example 7
53:06 - Example 8

[pschymit]

In Example 4, I see a possible very minor simplification : (-1)^(n-6) can be expressed as (-1)^n, as the alternation between -1 and 1 remains the same. Sorry for these little points; I’m greatly enjoying this refresher for me, just encountering your terrific videos years after they were published!

[pschymit]

A minor correction (I think) with example 3: you stated that 1/(1+x^2) = 1/(1-[-x]^2) . I think that should read 1/(1+x^2) = 1/(1-[-(x^2)])

[nicolasazzolini]

When integrating a series, does the starting value of n switch? (when differentiating it goes from 0 to 1, but I wasn't sure if it worked the same for integrals)

[jasonmantri]

how do you know whether to differentiate or integrate in these series?