Are there power series which diverge everywhere? If so, exhibit one. If not explain why it cannot happen.

Question

Are there power series which diverge everywhere? If so, exhibit one. If not explain why it cannot

happen.

Answer 1

Any power series converges at its center. So, a given power series can't diverge for all real numbers.

Answer 2

Try the sum of the product of n! and xⁿ.