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anonymous
 3 years ago
Power series problem:
n!(2x1)^n
from 1 to infinity
anonymous
 3 years ago
Power series problem: n!(2x1)^n from 1 to infinity

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abb0t
 3 years ago
Best ResponseYou've already chosen the best response.0are you being asked to determine whether the power series converges or not?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0when simplified i got absolute value of (2x1) Lim n>00 n+1

abb0t
 3 years ago
Best ResponseYou've already chosen the best response.0use ratio test. works about 7580% of the time for most power series. and best test to use when you have factorials.

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0i did that, but not sure if i am right..

abb0t
 3 years ago
Best ResponseYou've already chosen the best response.0\[L = \lim_{n \rightarrow \infty}\left \frac{ a_{n+1} }{ a_n } \right\] if L <1 absolutely convergent, and thus convergent L > 1 is divergent L = 1 use different test. but it SHOULD work.

abb0t
 3 years ago
Best ResponseYou've already chosen the best response.0\[L = \lim_{n \rightarrow \infty}\left \frac{ (n+1)! }{ (2x1)^{n+1} } \times \frac{(2x1)^n}{n!} \right = \lim_{n \rightarrow \infty }\left \frac{ n(n+1)}{ (2x1)(2x1)^n } \ \times \frac{ (2x1)^n }{ n } \right\]

abb0t
 3 years ago
Best ResponseYou've already chosen the best response.0I'm sure you can finish it from here.

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0so when simplified should be (n+1)(2x1) right?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0so i pull the 2x1 out and im left with the Lim n>00 of n+1 ?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0Your series diverges everywhere except for x =1/2. You can do it by the divergence test, the nth terms does not go to zero, then the series diverges

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0hey Elias, how did you figure that out?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0It is a power series. if \( x\ne \frac 1 2\), then n!(2x1)^n goes to infinity with n, so the nthterm does not go to zero, so the series is divergent.
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