Question

Determine the radius of convergence and interval of convergence for the power series: ∞∑n=0 (x−7)^n+1/n^n.

(a) 0, −1<x<1

(b) ∞, −∞<x<∞

(c) 1, −2<x<2

(d) 2, −1<x<1

The question was asked during an interview.

Question is from Discrete Probability topic in section Discrete Probability of Discrete Mathematics

Answer 1

The correct choice is (b) ∞, −∞<x<∞

Explanation: So, L=limn→∞∣(x−7)^n+1/n^n∣

L=limn→∞∣x−7/n∣

L=|x−7|limn→∞1/n=0

So, since L=0<1 any of the value of x, this power series will converge for every x. In these cases, the radius of convergence is R=∞ and interval of convergence is −∞<x<∞.