Question

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

• A. $[-3,3]$
• B. $[-6,6)$
• C. $(-6,6)$
• D. $(-3,3)$

Answer 1

The correct option is B $[-6,6)$

${\sum }_{n=1}^{\infty }\frac{x^n}{n{6}^n}$

$={lim}_{n\to \infty }\frac{x^n}{(n+1)(6^n+1)}\ast \frac{n\left(6^n\right)}{x^n}$

$={lim}_{n\to \infty }\frac{x\ast n}{(n+1)6}$

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

$\frac{x}{6}<1$ (series converges)

x<6

$\frac{x}{6}>-1$ (series diverges)

x>-6

$x\in [-6,6)$