42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)(1 − cos(1 / k))²

Verified step by step guidance

1. First, write down the series explicitly: \( \sum_{k=1}^{\infty} \left(1 - \cos\left(\frac{1}{k}\right)\right)^2 \). Our goal is to determine whether this infinite series converges or diverges.
2. Recall that for large \(k\), \( \frac{1}{k} \) is close to zero. We can use the Taylor series expansion of \( \cos x \) around \( x = 0 \) to approximate \( \cos\left(\frac{1}{k}\right) \). The expansion is \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \).
3. Substitute \( x = \frac{1}{k} \) into the expansion to approximate \( 1 - \cos\left(\frac{1}{k}\right) \). This gives \( 1 - \cos\left(\frac{1}{k}\right) \approx \frac{1}{2k^2} - \frac{1}{24k^4} + \cdots \).
4. Since the series term is \( \left(1 - \cos\left(\frac{1}{k}\right)\right)^2 \), square the approximation to find the leading term behavior. The dominant term will be proportional to \( \frac{1}{k^4} \) because \( \left(\frac{1}{2k^2}\right)^2 = \frac{1}{4k^4} \).
5. Compare the series \( \sum \left(1 - \cos\left(\frac{1}{k}\right)\right)^2 \) to the p-series \( \sum \frac{1}{k^4} \). Since \( p = 4 > 1 \), the p-series converges, so by the Comparison Test, the original series converges as well.

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Determining convergence involves analyzing the behavior of the terms as the index grows large, ensuring the sum does not diverge to infinity or oscillate indefinitely.

Comparison and Limit Comparison Tests

These tests compare the given series to a known benchmark series to determine convergence. The Comparison Test uses inequalities, while the Limit Comparison Test uses the limit of the ratio of terms, helping to infer convergence or divergence by relating to simpler, well-understood series.

Asymptotic Behavior and Series Term Approximation

Analyzing the behavior of terms for large indices often involves approximations like Taylor expansions. For example, approximating cos(1/k) for large k helps simplify terms to known forms, enabling the use of standard convergence tests on the approximated series.