Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
Determine if the series ∑n=1∞−1n+1n+1 diverges, converges absolutely, or converges conditionally.
If it converges conditionally, determine if it also converge absolutely.
The given series is alternating, and limn→∞1/n+1 = 0, and the sequence 1/n+1 is monotone decreasing, with limit zero.
Hence, by the Leibniz test, the series converges conditionally.
However, it does not converge absolutely, as can be seen via the Limit-Comparison test, using the divergent p-series Σ 1/n (p=1/2<1):
limn→∞1/n+11/n = limn→∞nn+1=1
By part (1) of the Limit-Comparison test, both series will converge or diverge, and since the p-series diverges, so also does the given series if all its terms are positive.
<< Previous Example Section 8.3
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)