Question

I need to find radius of convergence of: \[\sum_{n=1}^{\infty}\frac{ (-1)^{n+1} }{ 2n }x^n\]
Can this be done by? \[\lim_{n \rightarrow \infty}\left| \frac{ a_{n+1} }{ a_n } \right|\]

Answer 1

Where \[a_n=\frac{ (-1)^{n+1} }{ 2n }\] right?

Answer 2

It is bugging for me, dont know if it is for you. But to easier see the question.
I need to find radius of convergence of: \[\sum_{n=1}^{\infty}\frac{ (-1)^{n+1} }{ 2n }x^n\]
Can this be done by? \[\lim_{n \rightarrow \infty}\left| \frac{ a_{n+1} }{ a_n } \right|\]

Answer 3

yes, use the ratio test

Answer 4

you get that the limit is 1 pretty much in your head

Answer 5

and since you are working with the absolute value, you can safely ignore the \((-1)^n\) part

Answer 6

Is the ratio test \[\lim_{n \rightarrow \infty}\left| \frac{ a_n }{ a_{n+1} } \right|\]
or the 1 i posted before?

Answer 7

the first one

Answer 8

Thats what I thought, but then I found the other formular here. https://en.wikipedia.org/wiki/Radius_of_convergence#Theoretical_radius

Answer 9

but it is only \[\lim_{n\to \infty}\frac{2n+2}{2n}\]

Answer 10

oh sorry, you are right i had it upside down

Answer 11

Yeap, got it. Thanks for the help :)