Question

Power Series of Sum (n / n^2+1) X^n For radius and interval of convergence

Answer 1

first you must add n and n you get 2m
then you divide and get fsafsafsdfgsfds

Answer 2

So far the only thing i know is devideing the equation with n+1 substituted with n divided by the original but i get stuck solving
\[\lim_{n \rightarrow \inf} (n+1/(n+1)^2)x^
(n+1) / (n/n^2+1 *x^n)\] the limit at that point

Answer 3

Ratio test:
\[\lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right|<1\]
So:
\[\lim_{n \rightarrow \infty}\frac{(n+1)x^{n+1}(n^2+1)}{((n+1)^2+1)(nx^n)}=\lim_{n \rightarrow \infty}\frac{(n^3+n^2+n+1)x}{n^3+2n^2+2n}=x \implies -1<x<1\]
So the open interval is that, now check the endpoints. Plugging in 1 gives us a divergent series (from mental math, check) and -1 gives us a convergent series (I believe).