anonymous
  • anonymous
I need to find radius of convergence of: \[\sum_{n=1}^{\infty}\frac{ (-1)^{n+1} }{ 2n }x^n\] Can this be done by? \[\lim_{n \rightarrow \infty}\left| \frac{ a_{n+1} }{ a_n } \right|\]
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
Where \[a_n=\frac{ (-1)^{n+1} }{ 2n }\] right?
anonymous
  • anonymous
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? \[\lim_{n \rightarrow \infty}\left| \frac{ a_{n+1} }{ a_n } \right|\]
anonymous
  • anonymous
yes, use the ratio test

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anonymous
  • anonymous
you get that the limit is 1 pretty much in your head
anonymous
  • anonymous
and since you are working with the absolute value, you can safely ignore the \((-1)^n\) part
anonymous
  • anonymous
Is the ratio test \[\lim_{n \rightarrow \infty}\left| \frac{ a_n }{ a_{n+1} } \right|\] or the 1 i posted before?
anonymous
  • anonymous
the first one
anonymous
  • anonymous
Thats what I thought, but then I found the other formular here. https://en.wikipedia.org/wiki/Radius_of_convergence#Theoretical_radius
anonymous
  • anonymous
but it is only \[\lim_{n\to \infty}\frac{2n+2}{2n}\]
anonymous
  • anonymous
oh sorry, you are right i had it upside down
anonymous
  • anonymous
Yeap, got it. Thanks for the help :)

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