Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
Determine if the series ∑n=1∞12+n diverges, converges absolutely, or converges conditionally.
If it converges conditionally, determine if it also converge absolutely.
Although 12+n→0 as n→∞, n<n so that the denominator is smaller than n, the denominator of the harmonic series, and hence, the terms of this series will be greater than the terms of the divergent harmonic series. This observation suggests the Limit-Comparison test, with the harmonic series as the comparison series. The relevant calculation is then
By part (3) of the Limit-Comparison test
, if the comparison series diverges, so also the given series diverges.
Alternatively, part (1) of the Limit-Comparison test with the divergent p-series Σ 1/n (p=1/2<1) as the comparison series also establishes the divergence of the given series because limn→∞12+n1/n = 1.
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