Absolute convergence
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Absolute convergence

[From: ] [author: ] [Date: 11-11-18] [Hit: ]
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Is the following series absolutely convergent Summation n=1 to infinity ((-1)^n-1)(e^(1/n))/n

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We determine whether Σ(n = 1 to ∞) |(-1)^(n-1) e^(1/n) / n|
= Σ(n = 1 to ∞) e^(1/n) / n is convergent.

Using the limit comparison test, note that
lim(n→∞) [e^(1/n) / n] / (1/n) = lim(n→∞) e^(1/n) = e^0 = 1.

Since Σ(n = 1 to ∞) 1/n is the divergent harmonic series, we conclude that
Σ(n = 1 to ∞) e^(1/n) / n must also diverge.

I hope this helps!
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keywords: convergence,Absolute,Absolute convergence
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