Question

# Find the radius of convergence and interval of convergence of the series. sum_{n=1}^inftyfrac{x^n}{n5^n}

Series
Find the radius of convergence and interval of convergence of the series.
$$\sum_{n=1}^\infty\frac{x^n}{n5^n}$$

2021-03-12

Consider the function as,
$$a_n=\frac{x^n}{n5^n}$$
So it implies that,
$$a_{n+1}=\frac{x^{n+1}}{(n+1)5^{n+1}}$$
Apply the ratio test, for a converging series,
$$\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|<1$$
Substitute the values and simplify,
$$\Rightarrow\lim_{n\to\infty}|\frac{\frac{x^{n+1}}{(n+1)\cdot5^{n+1}}}{\frac{x^n}{n\cdot5^n}}|<1$$
$$\Rightarrow\lim_{n\to\infty}|\frac{x^{n+1}}{(n+1)\cdot5^{n+1}}\times\frac{n\cdot5^n}{x^n}|<1$$
$$\Rightarrow\lim_{n\to\infty}|\frac{x}{(n+1)\cdot5}\times n|<1$$
$$\Rightarrow\lim_{n\to\infty}|\frac{x}{\frac{1}{n}\cdot(n+1)\cdot5}|<1$$
Simplify the terms further
$$\Rightarrow\lim_{n\to\infty}|\frac{x}{(1+\frac{1}{n})\cdot5}|<1$$
$$\Rightarrow|\frac{x}{(1+0)\cdot5}|<1$$
$$\Rightarrow|x|<5$$
Thus, the radius of convergence of the given series is 5 units.
Consider the inequality as,
$$|x|<5$$
$$\Rightarrow-5$$
or $$x\in(-5,5)$$
Thus, the interval of convergence is (−5, 5).