Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.∑ (k = 1 to ∞) k / eᵏ

Accepted Answer

Identify the series given: $$ \sum_{k=1}^{\infty} \frac{k}{e^{k}} . $$Notice that the terms involve $$ k  in $$the numerator and an exponential $$ e^{k}  in $$the denominator. Consider the behavior of the terms $$ a_k = \frac{k}{e^{k}}  as$$ $$ k 	o \infty . $$Since the denominator grows exponentially and the numerator grows linearly, the terms $$ a_k $$ approach zero, which is a necessary condition for convergence. Apply the Divergence Test first: check if $$ \lim_{k 	o \infty} a_k 
eq 0 . If $$the limit is not zero, the series diverges. Here, the limit is zero, so the Divergence Test is inconclusive. Use the Integral Test or compare with a known convergent series. Since $$ e^{k} $$ grows faster than any polynomial, compare $$ \frac{k}{e^{k}}  to$$ $$ \frac{1}{e^{k/2}}  or $$recognize it as a series with terms decreasing exponentially. Conclude that the series converges by comparison to a convergent geometric series or by applying the Integral Test to the function $$ f(x) = \frac{x}{e^{x}} $$, which is positive, continuous, and decreasing for $$ x \geq 1 .$$

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) k / eᵏ

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Key Concepts

Divergence Test

Integral Test

p-series Test