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!
= 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!