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)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

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)