Question

Find whether the following series are convergent or divergent:
$\frac{x}{1.2}+\frac{x^2}{2.3}+\frac{x^3}{3.4}+\frac{x^4}{4.5}+...$

Answer 1

To find whether the following series is convergent or divergent

Given series is

$\frac{x}{1.2}+\frac{x^2}{2.3}+\frac{x^3}{3.4}+\frac{x^4}{4.5}+......$

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

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

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

On simplifying this, we get

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

Finally on applying limit, we have

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

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

For convergence,

$L<1$

$\left|x\right|<1$

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

i.e. series converges for $x\le 1$ and diverges for $x>1$