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)

Verified step by step guidance

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

Key Concepts

Telescoping Series

A telescoping series is a series where many terms cancel out when expanded, leaving only a few terms from the beginning and end. This simplification helps in evaluating the sum of infinite series by reducing complex expressions to simpler forms.

Properties of the Inverse Tangent Function (arctan)

The inverse tangent function, arctan(x), is continuous and monotonic, with known limits as x approaches infinity or negative infinity. Understanding its behavior helps in evaluating limits of terms like arctan(k+2) - arctan(k) as k grows large.

Convergence and Divergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit; otherwise, it diverges. Determining convergence often involves analyzing term behavior and applying tests or simplifications like telescoping.