38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.
39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 1)

Verified step by step guidance

1. First, recognize the general term of the series: \(a_k = \frac{1}{k^2 + 2k + 1}\). Notice that the denominator can be factored as \((k + 1)^2\).
2. Rewrite the series as \(\sum_{k=1}^\infty \frac{1}{(k+1)^2}\). This is similar to the p-series \(\sum \frac{1}{k^p}\) with \(p=2\).
3. Use the Comparison Test by comparing \(a_k\) to \(b_k = \frac{1}{k^2}\). Since \(\frac{1}{(k+1)^2} < \frac{1}{k^2}\) for all \(k\), and we know \(\sum \frac{1}{k^2}\) converges (p-series with \(p=2 > 1\)), the original series converges by comparison.
4. Alternatively, apply the Limit Comparison Test with \(b_k = \frac{1}{k^2}\). Compute the limit \(L = \lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} \frac{1/(k+1)^2}{1/k^2} = \lim_{k \to \infty} \left( \frac{k}{k+1} \right)^2\).
5. Since \(L = 1\) (a finite positive number), both series either converge or diverge together. Because \(\sum \frac{1}{k^2}\) converges, the original series converges as well.

Key Concepts

Comparison Test

The Comparison Test determines the convergence of a series by comparing it to a second series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges. Conversely, if the terms are larger than those of a divergent series, it diverges.

Limit Comparison Test

The Limit Comparison Test compares two series by taking the limit of the ratio of their terms. If the limit is a positive finite number, both series either converge or diverge together. This test is useful when direct comparison is difficult but the series have similar term behavior.

Recognizing and Simplifying Series Terms

Simplifying the general term of a series can reveal its form and help identify a comparable series. For example, rewriting 1/(k² + 2k + 1) as 1/(k+1)² shows it resembles a p-series with p=2, which is known to converge. Recognizing such forms aids in applying convergence tests effectively.