Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 1 to ∞) (cos(1 / k) – cos(1 / (k + 1)))

Accepted Answer

Recognize that the given series is $$ \sum_{k=1}^{\infty} \left( \cos\left(\frac{1}{k}ight) - \cos\left(\frac{1}{k+1}ight) ight) $$, which is a telescoping series because each term involves the difference of cosine values at consecutive indices. Write out the first few partial sums explicitly to observe the telescoping pattern: $$ S_n = \sum_{k=1}^n \left( \cos\left(\frac{1}{k}ight) - \cos\left(\frac{1}{k+1}ight) ight) = \left( \cos(1) - \cos\left(\frac{1}{2}ight) ight) + \left( \cos\left(\frac{1}{2}ight) - \cos\left(\frac{1}{3}ight) ight) + \cdots + \left( \cos\left(\frac{1}{n}ight) - \cos\left(\frac{1}{n+1}ight) ight) . $$Notice that most terms cancel out in the sum, leaving only the first term of the first cosine and the last term of the last cosine: $$ S_n = \cos(1) - \cos\left(\frac{1}{n+1}ight) . To $$determine convergence, analyze the limit of the partial sums as $$ n 	o \infty $$: $$ \lim_{n 	o \infty} S_n = \lim_{n 	o \infty} \left( \cos(1) - \cos\left(\frac{1}{n+1}ight) ight) . $$Since $$ \lim_{x 	o 0} \cos(x) = 1 $$, evaluate the limit to find $$ \lim_{n 	o \infty} S_n = \cos(1) - 1 . $$This limit exists and is finite, so the series converges by the telescoping series test.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)(cos(1 / k) – cos(1 / (k + 1)))

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Key Concepts

Telescoping Series

Limit of a Sequence

Convergence Tests for Series

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 1 to ∞)(cos(1 / k) – cos(1 / (k + 1)))