Question

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.∑ (from k = 0 to ∞) (tan⁻¹(k + 2) − tan⁻¹k)

Accepted Answer

Recognize that the series is given by the sum from k = 0 to infinity of the terms $$ 	an^{-1}(k + 2) - 	an^{-1}(k) . $$This is a telescoping series because each term is a difference of inverse tangent functions evaluated at consecutive points separated by 2. Write out the first few partial sums explicitly to observe the telescoping pattern: $$ S_n = \sum_{k=0}^n \left( 	an^{-1}(k + 2) - 	an^{-1}(k) ight) = (	an^{-1}(2) - 	an^{-1}(0)) + (	an^{-1}(3) - 	an^{-1}(1)) + \cdots + (	an^{-1}(n+2) - 	an^{-1}(n)) . $$Group the terms in the partial sum to see which terms cancel out. Notice that many intermediate terms will cancel, leaving only a few terms from the beginning and the end of the sum. Express the partial sum $$ S_n  in $$terms of the remaining terms after cancellation. Typically, for telescoping sums of this form, $$ S_n = 	an^{-1}(n+1) + 	an^{-1}(n+2) - 	an^{-1}(0) - 	an^{-1}(1)  or a $$similar expression depending on the exact cancellation pattern. Evaluate the limit of the partial sums as $$ n 	o \infty  to $$determine if the series converges. Use the fact that $$ \lim_{x 	o \infty} 	an^{-1}(x) = \frac{\pi}{2}  to $$find the limit of $$ S_n . If $$the limit exists and is finite, the series converges to that value; otherwise, it diverges.

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 0 to ∞)(tan⁻¹(k + 2) − tan⁻¹k)

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Key Concepts

Telescoping Series

Properties of the Inverse Tangent Function (arctan)

Convergence and Divergence of Infinite Series

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.∑ (from k = 0 to ∞)(tan⁻¹(k + 2) − tan⁻¹k)