Question

Match each of the power series with its interval of convergence.

Answer 1

Is this right? i only get one try so I wanted to be sure i got them ><

Answer 2

Ill work them myself real quick to see

Answer 3

aww thanks @Concentrationalizing i can post my work if that's easier

Answer 4

I have better luck working it myself. Ive noticed that I can miss something a student does wrong if I just scan their work.

Answer 5

haha sounds good :)

Answer 6

1st one is fine.

Answer 7

kk :)

Answer 8

3 and 4 need to be flip-flopped

Answer 9

Ill show the work

Answer 10

kk thanks :)

Answer 11

so it is C,B,D,A

Answer 12

And if i could see ur work i'd appreciate that :)

Answer 13

Via root test on #3
\[\lim_{n \rightarrow \infty} \sqrt[n]{\left| \frac{ (x-9)^{n} }{ 9^{n} } \right|}\]
\[= \left| x-9 \right|\cdot \lim_{n \rightarrow \infty}\frac{ 1 }{ 9 } = \frac{ 1 }{ 9 }\left| x-9 \right|\]
All the n's cancel and this is your limit. The conditions for root test are the same as for ratio test, we need to be less than 1. Thus we have:
\[\frac{ 1 }{ 9 }\left| x-9 \right| < 1 \implies \left| x-9 \right| < 9\] which would give you the (0,18) result (since its multiple choice, I assume we dont need to actually check the endpoints)

Answer 14

i will not butt in and let @Concentrationalizing finish, but i am willing to bet you can guess at least two of these doing no work
now i will go away

Answer 15

Via ratio test on #4
\[\lim_{n \rightarrow \infty}\left| \frac{ (x-9)^{n+1} }{ (n+1)!9^{n+1} }\cdot \frac{ 9^{n}n! }{ (x-9)^{n} } \right|\]
\[= \left| x-9 \right|\lim_{n \rightarrow\infty}\left| \frac{ 9^{n}n! }{ 9(n+1)9^{n}n! } \right|\]
\[= \left| x-9 \right|\lim_{n \rightarrow \infty}\left| \frac{ 1 }{ 9(n+1) } \right| = 0\]
So all values of x work since we got a result of 0

Answer 16

I could guess them, but I'm not comfortable enough doing that yet, I'd rather just do the work and make sure I'm correct, lol @satellite73

Anyway, normally these aren't multiple choice x_x

Answer 17

haha thanks :)

Answer 18

No problem :)