Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.2 Series - 11.2 Exercises - Page 756: 34


The series diverges since it is the sum of a converging and diverging geometric series

Work Step by Step

We can split the series $\sum\limits^{\infty}_{n=1} \frac{2^n + 4^n}{e^n}$ into the sum of two geometric series $\sum\limits^{\infty}_{n=1} \frac{2^n}{e^n} +\frac{4^n}{e^n} $ We can rewrite the two series as $\sum\limits^{\infty}_{n=1} (\frac{2}{e})^n +(\frac{4}{e})^n $ and see that both are geometric series. Recall that a geometric series converges if the common ratio $r$ satisfies $$|r| \lt 1$$ The first series has a common ratio $\frac{2}{e} \lt 1$ so the series converges. However, the second series has a common ratio $\frac{4}{e} \gt 1$ so the series diverges. The sum of a converging series and a diverging series diverges, thus this series diverges.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.