Question

For the power series of the form, \(∑_{i=0}^∞ a_i z^i,\) which one of the following may not be true?

(a) The series converges only for z=0

(b) The series converges absolutely for all z

(c) The series converges absolutely for all z in some finite open interval (-R, R) and diverges if z<-R or z>R

(d) At the points z=R and z=-R, the series will diverge

I have been asked this question during an interview.

My question is from Generalized Mean Value Theorem in portion Differential Calculus of Engineering Mathematics

Answer 1

Correct choice is (d) At the points z=R and z=-R, the series will diverge

Explanation: A power series in a variable z is in the form of an infinite sum as given, i.e., \(∑_{i=0}^∞ a_i z^i,\), where ai are integers, real numbers, complex numbers, or any other quantities of a given type.

For any power series, one of the following is true:

i. The series converges only for z=0.

ii. The series converges absolutely for all z.

iii. The series converges absolutely for all z in some finite open interval (-R, R) and diverges if z<-R or z>R. At the points z=R and z=-R, the series may converge absolutely, converge conditionally, or diverge.