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.
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