Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.{Use of Tech} ∑ (k = 1 to ∞) 1 / (4 ln k)

Accepted Answer

Identify the given series: $$ \sum_{k=1}^{\infty} \frac{1}{4 \ln k} . $$Note that the term for $$k=1 is $$undefined since $$\ln 1 = 0$$, so consider the series starting from $$k=2$$ instead. Recall that the Comparison Test and Limit Comparison Test are used to determine convergence by comparing the given series to a known benchmark series. Since the terms involve $$ \frac{1}{\ln k} $$, consider comparing it to the harmonic series $$ \sum \frac{1}{k} $$, which is divergent. Set up the Limit Comparison Test by defining $$a_k = \frac{1}{4 \ln k} $$ and $$b_k = \frac{1}{k} . $$Compute the limit $$ L = \lim_{k 	o \infty} \frac{a_k}{b_k} = \lim_{k 	o \infty} \frac{1/(4 \ln k)}{1/k} = \lim_{k 	o \infty} \frac{k}{4 \ln k} . $$Analyze the limit $$ L = \lim_{k 	o \infty} \frac{k}{4 \ln k} . $$Since $$k$$ grows faster than $$\ln k$$, this limit tends to infinity, which means the Limit Comparison Test with $$b_k = \frac{1}{k} is $$inconclusive. Try a different comparison series, such as $$b_k = \frac{1}{k^p} $$ for some $$p \u003e 0$$, or consider the integral test for $$ \int \frac{1}{\ln x} dx  to $$analyze convergence. The integral test can help determine if the series converges or diverges by examining the behavior of the integral of the function $$f(x) = \frac{1}{4 \ln x} $$ for $$x \geq 2.$$

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
{Use of Tech} ∑ (k = 1 to ∞) 1 / (4lnk)

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Key Concepts

Comparison Test

Limit Comparison Test

Behavior of the Harmonic Series and Logarithmic Terms

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.{Use of Tech} ∑ (k = 1 to ∞) 1 / (4lnk)