Question

Find whether the following series is convergent or divergent:
$1+\frac{2}{5}x+\frac{6}{9}{x}^2+\frac{14}{17}{x}^3+...\frac{2^n-1}{2^n+1}{x}^{n-1}+...$

Answer 1

To find whether the following series is convergent or divergent

$1+\frac{2}{5}x+\frac{6}{9}{x}^2+\frac{14}{17}{x}^3+.....+\frac{2^n-1}{2^n+1}{x}^{n-1}$

Here,

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

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

Also,

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

$\Rightarrow \underset{n\to \infty }{lim}\frac{a_n}{a_{n-1}}=\underset{n\to \infty }{lim}\frac{\frac{2^n-1}{2^n+1}{x}^n}{\frac{2^{n-1}-1}{2^{n-1}+1}{x}^{n-1}}$

On applying limit, we have

$\Rightarrow {lim}_{n\to \infty }\frac{a_n}{a_{n-1}}=x$

Hence, if $x>1$, then the series is divergent

and if $x<1$, then the series is convergent

If $x=1$ then $\underset{n\to \infty }{lim}{a}_n=1$

Hence, the series is divergent for $x=1$.