Mathematical Methods in the Physical Sciences · Problem 6.17

INFINITE SERIES, POWER SERIES — Mary Boas

Mathematics — Methods · integral-test

Problem

Use the integral test to show that

\displaystyle\sum_{n=0}^{\infty} e^{-n^2}

converges. Hint: Although you cannot evaluate the integral, you can show that it is finite (which is all that is necessary) by comparing it with

\int^\infty e^{-n}\,dn