Question

Consider the power series $\sum _{n=1}^{\infty }\frac{x^n}{n\cdot 6^n}$ then find the radius of the convergence.

• A. $3$
• B. $6$
• C. $8$
• D. $2$

Answer 1

The correct option is B $6$

Given series is $\sum _{n=1}^{\infty }\frac{x^n}{n.6^n}$

Using the ratio test

$\left|\frac{a_{n+1}}{a_n}\right|=\left|\frac{\frac{x^{n+1}}{(n+1)6^{n+1}}}{\frac{x^n}{n.6^n}}\right|$

$\left|\frac{a_{n+1}}{a_n}\right|=\left|\frac{xn}{6(n+1)}\right|$

$\underset{n\to \infty }{lim}\left|\frac{a_{n+1}}{a_n}\right|=\left|\frac{x}{6}\right|\underset{n\to \infty }{lim}\frac{n}{n+1}$

$\underset{n\to \infty }{lim}\left|\frac{a_{n+1}}{a_n}\right|=\left|\frac{x}{6}\right|$

For convergence

$\left|\frac{x}{6}\right|<1-1<\frac{x}{6}<1-6<x<6$

Therefore, radius of convergence $=\frac{6-(-6)}{2}=6$.