Question

Find whether the following series is convergent or divergent:
$1+\frac{x}{2}+\frac{x^2}{5}+\frac{x^3}{10}+....+\frac{x^n}{n^2+1}+...$

Answer 1

To find whether the following series is convergent or divergent

Given series is

$1+\frac{x}{2}+\frac{x^2}{5}+\frac{x^3}{10}+......+\frac{x^n}{n^2+1}+....$

$a_n=\frac{x^n}{n^2+1}$

According to Ratio test

If there exists an $N$ so that $n\ge N$

If $L<1$, then the series converges

If $L>1$, then the series diverges

If $L=1$, then the Ratio test is inconclusive

where, $L=\underset{n\to \infty }{lim}\left|\frac{a_{n+1}}{a_n}\right|$

On applying ratio test, we get

$L=\underset{n\to \infty }{lim}\left|\frac{a_{n+1}}{a_n}\right|$

$\Rightarrow L=\underset{n\to \infty }{lim}\left|\frac{\frac{x^{n+1}}{(n+1{)}^2+1}}{\frac{x^n}{n^2+1}}\right|$

$\Rightarrow L=\underset{n\to \infty }{lim}\left|\frac{x(n^2+1)}{(n+1{)}^2+1}\right|$

$\Rightarrow L=\underset{n\to \infty }{lim}\left|\frac{x(n^2+1)}{n^2+2n+2}\right|$

$\Rightarrow L=\underset{n\to \infty }{lim}\left|\frac{x(1+\frac{1}{n^2})}{1+\frac{2}{n}+\frac{2}{n^2}}\right|$

On applying limit, we have

$\Rightarrow L=\left|x\right|$

For convergence, $L<1$

i.e. $\left|x\right|<1$

i.e. $-1\le x\le 1$

Therefore, given series converges for $-1\le x\le 1$

And diverges for $x>1$