Question

What is the radius of convergence and interval of convergence for the power series ∞∑n=0m!(2x-1)^m?

(a) 3, 12

(b) 1, 0.87

(c) 2, 5.4

(d) 0, 1/2

I got this question during an interview.

This question is from Discrete Probability in section Discrete Probability of Discrete Mathematics

Answer 1

Right choice is (d) 0, 1/2

Explanation: Suppose, L=limn→∞ |(m+1)!(2x+1)^m+1/m!(2x+1)^m|

= limm→∞∣(m+1)m!(2x-1)/m!|

= |2x-1|limm→∞(m+1)

So, this power series will only converge if x=1/2. We know that every power series will converge for x=a and in this case a=1/2. Remember that we get a from (x−a)^n. In this case, the radius of convergence is R=0 and the interval of convergence is x=1/2.