Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 1 to ∞) (1 / √(k + 2) – 1 / √k)

Accepted Answer

Recognize that the series is given by $$ \sum_{k=1}^{\infty} \left( \frac{1}{\sqrt{k+2}} - \frac{1}{\sqrt{k}} ight) . $$This is a series whose terms are differences of two expressions involving square roots. Rewrite the general term to see if the series is telescoping. Notice that each term looks like $$ a_k = \frac{1}{\sqrt{k+2}} - \frac{1}{\sqrt{k}} . $$Try to express the partial sums $$ S_n = \sum_{k=1}^n a_k  to $$identify cancellation patterns. Write out the first few terms of the partial sum explicitly: $$ S_n = \left( \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{1}} ight) + \left( \frac{1}{\sqrt{4}} - \frac{1}{\sqrt{2}} ight) + \cdots + \left( \frac{1}{\sqrt{n+2}} - \frac{1}{\sqrt{n}} ight) . $$Group the positive and negative terms to observe which terms cancel out. After cancellation, identify the remaining terms in $$ S_n . $$Typically, in telescoping series, most intermediate terms cancel, leaving only a few terms from the beginning and end of the sequence. Analyze the limit of the partial sums $$ S_n  as$$ $$ n 	o \infty . If $$the limit exists and is finite, the series converges; otherwise, it diverges.

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

∑ (from k = 1 to ∞)(1 / √(k + 2) – 1 / √k)

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Key Concepts

Telescoping Series

Convergence of Infinite Series

Comparison and Limit Comparison Tests

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 1 to ∞)(1 / √(k + 2) – 1 / √k)