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

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

Mathematics
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Is this right? i only get one try so I wanted to be sure i got them ><
Ill work them myself real quick to see
aww thanks @Concentrationalizing i can post my work if that's easier

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I have better luck working it myself. Ive noticed that I can miss something a student does wrong if I just scan their work.
haha sounds good :)
1st one is fine.
kk :)
3 and 4 need to be flip-flopped
Ill show the work
kk thanks :)
so it is C,B,D,A
And if i could see ur work i'd appreciate that :)
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)
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
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
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
haha thanks :)
No problem :)

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