Question

Find whether the following series are convergent or divergent:
$1+3x+5x^2+7x^3+9x^4+.....$

Answer 1

To find whether the following series are convergent or divergent

$1+3x+5x^2+7x^3+9x^4+....$

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

$a_{n-1}=(2n-3)x^{n-2}$

Also,

$\frac{a_n}{a_{n-1}}=\frac{(2n-1)x^{n-1}}{(2n-3)x^{n-2}}$

$\frac{a_n}{a_{n-1}}=\frac{(2n-1)x}{(2n-3)}$

$\Rightarrow \underset{n\to \infty }{lim}\frac{a_n}{a_{n-1}}=\underset{n\to \infty }{lim}\frac{(2n-1)x}{(2n-3)}$

$\Rightarrow \underset{n\to \infty }{lim}\frac{a_n}{a_{n-1}}=\underset{n\to \infty }{lim}\frac{(2-\frac{1}{n})x}{(2-\frac{3}{n})}$

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

Hence, if $x<1$, the series is convergent

If $x>1$, the series is divergent

If $x=1$, then the series is $1+3+5+7+9+...........$

Therefore, it is divergent for $x=1$