2.17.32 - Mary L Boas.

Question

Use a series you know to show that $\sum_{n=0}^\infty\frac{(1+i\pi)^n}{n!}=-e.$

It does look like $e^x$ at first, but i'm not sure how to proceed. Any tips?

Answer 1

Yep, it is $e^{1+i\pi}=e(\cos\pi + i\sin\pi)=e(-1)=-e$.

Answer 2

A hint: $$ \ln(-e) = \ln(e) +\ln(-1) = 1 + \ln(-1) $$ You can use Euler equation to find that $$ \mathrm{e}^{i\pi} $$ So you have the argument in the summand.