1. Determine the radius and the interval of convergence of...

1. Determine the radius and the interval of convergence of the following power series.\\ (a) $\sum_{n=0}^{\infty} (-1)^n \frac{(x - 3)^n}{2n + 1}$\\ (b) $\sum_{n=0}^{\infty} \frac{3^n n^2 (x - 8)^n}{(n + 1)!}$

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Transcript

00:01 Okay we want to find the radius of convergence and also the interval of convergence of some power series.

00:10 So here's the first one.

00:31 So let's use the ratio test.

00:34 It's always a good place to start.

00:37 We're looking for intervals.

00:46 So then we get, i get to factor out an x.

01:39 Then i got 2n plus 1 factorial over 2n factorial.

01:44 It gives me 2n plus 2, 2n plus 1.

01:53 Then i got n to the 2n over n plus 1 to the 2n plus 2.

02:05 So we got 2n plus 2, 2n plus 1.

02:20 Then this thing i can factor out an n plus 1 squared and then i have n over n plus 1 raised to the 2n power.

02:43 So this limit here is 1 over e squared.

02:52 So we get x over e squared and then we got the limit of this part is 4.

03:03 It's the highest power of n.

03:07 The numerator and denominator says 4 over 1.

03:11 So that is equal to, okay so that's 0 .54 times x.

03:19 So that means that 0 .54 x is between minus 1 and 1 or x is between minus 1 over 0 .54 is less than 1 over 0 .54.

03:47 Okay so what about at the endpoints? so at the endpoints, so this is 4 over e squared.

04:05 So you flip that over...

Question

1. Determine the radius and the interval of convergence of the following power series.\\ (a) $\sum_{n=0}^{\infty} (-1)^n \frac{(x - 3)^n}{2n + 1}$\\ (b) $\sum_{n=0}^{\infty} \frac{3^n n^2 (x - 8)^n}{(n + 1)!}$

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