Find the interval of convergence for the series:

Question

Quizplus · Accepted Answer

To find the interval of convergence of a power series, we use the ratio test. Let $a_n = (-1)^n\frac{2^{n+1}(x-3)^{n+1}}{3^{n+1}(n+1)!}$ be the $n$th term of the series. Then, we have:

$$\lim_{n	o\infty} \left|\frac{a_{n+1}}{a_n}ight| = \lim_{n	o\infty} \left| \frac{-2(x-3)}{3(n+2)} ight| = 0 < 1$$

Therefore, the series converges for all $x$, which means the interval of convergence is $(-\infty, \infty)$. The only choice that represents this interval is C.

Find the Interval of Convergence for the Series