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

Verified step by step guidance

1. Identify the series given: \( \sum_{k=1}^{\infty} \tan^{-1}\left( \frac{1}{\sqrt{k}} \right) \). We want to determine if this infinite series converges or diverges.
2. Recall that for large \( k \), \( \tan^{-1}(x) \) behaves approximately like \( x \) when \( x \) is close to zero. Since \( \frac{1}{\sqrt{k}} \to 0 \) as \( k \to \infty \), we can compare \( \tan^{-1}\left( \frac{1}{\sqrt{k}} \right) \) to \( \frac{1}{\sqrt{k}} \).
3. Use the Comparison Test or Limit Comparison Test by comparing the given series to the series \( \sum_{k=1}^{\infty} \frac{1}{\sqrt{k}} \), which is a p-series with \( p = \frac{1}{2} \). Recall that a p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \).
4. Since \( p = \frac{1}{2} < 1 \), the series \( \sum \frac{1}{\sqrt{k}} \) diverges. Therefore, if the terms of our original series behave like \( \frac{1}{\sqrt{k}} \) for large \( k \), the original series will also diverge by comparison.
5. Conclude that the series \( \sum_{k=1}^{\infty} \tan^{-1}\left( \frac{1}{\sqrt{k}} \right) \) diverges because its terms do not decrease fast enough to produce a convergent sum.

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.

Comparison Test and Limit Comparison Test

These tests help determine convergence by comparing the given series to a known benchmark series. If terms of the series behave similarly to a convergent or divergent series, the original series shares the same behavior, simplifying the analysis.

Behavior of arctan(x) for Small Arguments

For small values of x, arctan(x) approximates x because arctan(x) ~ x as x → 0. This approximation allows us to compare the given series terms tan⁻¹(1/√k) to simpler terms like 1/√k, aiding in applying convergence tests.