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I am slightly unsure about how to do the following question relating to the radius convergence (using specifically the ratio test). The power series is as follows:

$$\sum_{n=1}^{\infty}\frac{(2x+1)^n}{n^2}$$

Applying the ratio test, I got $$\lim_{n\rightarrow \infty} \left| \frac{(2x+1)n^2}{(n+1)^2}\right| < 1$$

I believe that at his point the, $n^2$ and $(n+1)^2$ should cancel out but I'm a bit puzzled as to why the radius of convergence is $1/2$ as given in the solutions.

Help is appreciated.

Thank you

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  • $\begingroup$ Yours power series is of the form $\sum \frac{2^{n}}{n^{2}}(x+1/2)^{n}$ clearly its radius of convergence is $\frac{1}{2}.$ $\endgroup$ – neelkanth Jun 28 '16 at 5:14
  • $\begingroup$ Oh, I see, you simply factored out the 2. $\endgroup$ – PutsandCalls Jun 28 '16 at 5:17

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