Answer to Question #116616 in Calculus for gly

Question

find the radius and interval of convergence of given series
∞
∑ (x+3)^n/2^n
n
=
0

Answer 1

Given series is $\sum_{n=0}^{\infin} u_n = \sum_{n=0}^{\infin} \frac{(x+3)^n}{2^n}.$

So, $u_n = \frac{(x+3)^n}{2^n}$

Now for convergence of series, $\lim_{n\to \infin} |\frac{u_{n+1}}{u_n}| < 1$

$\implies \lim_{n\to \infin} |\frac{(x+3)^{n+1}/2^{n+1}}{(x+3)^n/ 2^n}| < 1$

$\implies \lim_{n\to \infin} |\frac{x+3}{2}| < 1 \\ \implies |x+3| < 2$

So, radius of convergence of given series is 2.