Question

Is the series
$1^2+2^{2}x+3^2{x}^2+4^2{x}^3+.....$ convergent or divergent?

Answer 1

Given series is $1^2+2^{2}x+3^2{x}^2+4^2{x}^3+.....=\sum _{n=1}^{\infty }{n}^2{x}^{n-1}$

Let $u_n=n^2{x}^{n-1}$

Here,

$\underset{n\to \infty }{lim}\frac{u_n}{u_{n-1}}$

$=\underset{n\to \infty }{lim}\frac{n^2{x}^{n-1}}{{(n-1)}^2{x}^{n-2}}$

$=\underset{n\to \infty }{lim}\frac{n^{2}x}{{(n-1)}^2}$

$=x$
Hence if $x<1$ the series is convergent
If $x>1$ the series is divergent.
If $x=1$ the series becomes $1^2+2^2+3^2+4^2+.....$ and is obviously divergent.