Finding sumfunctions for power series

May 2010
98
16
Hello

I need to find the sumfunction and the radius of convergence + convergence range for the following:

\(\displaystyle \sum_{n=0}^{\infty} (2n+1)(x+7)^n\)

What i did was this:

\(\displaystyle \lim_{n\rightarrow\infty} \frac{2n+1}{2(n+1)+1}=\lim_{n\rightarrow\infty} \frac{2n+1}{2(n+3)}\) which converges to 1 for \(\displaystyle n\rightarrow\infty\) so radius of convergense is 1, but what is the convergence range?

Or am i totally off here?
 

Krizalid

MHF Hall of Honor
Mar 2007
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Santiago, Chile
\(\displaystyle |x+7|<1,\) now check the endpoints.
 
May 2010
98
16
\(\displaystyle |x+7|<1,\) now check the endpoints.
Yes thats what i came up with too, but i tried to calc it in maple using:

Sum((2*n+1)*((x+7)^n), n=0..infinity); simplify(value(%));

Which gave me \(\displaystyle \frac{x+8}{(x+6)^2}\)