Question

Find whether the following series are convergent or divergent:
$\frac{x}{1.2}+\frac{x^2}{3.4}+\frac{x^3}{5.6}+\frac{x^4}{7.8}+...$

Answer 1

To find whether the following series are convergent or divergent

Given series is

$\frac{x}{1.2}+\frac{x^2}{3.4}+\frac{x^3}{5.6}+\frac{x^4}{7.8}+......$

Here, $a_n=\frac{x^n}{2n.(2n-1)}$

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

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

On simplifying this, we get

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

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

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

For convergence, $L<1$

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

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

Thus, the series converges for $x\le 1$ and diverges for $x>1$.