42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)(7 + sin k) / k²

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} \frac{7 + \sin k}{k^2} \). We want to determine if this series converges or diverges.
2. Note that \( \sin k \) is bounded between -1 and 1, so the numerator \( 7 + \sin k \) is bounded between 6 and 8. This means the terms behave roughly like \( \frac{\text{constant}}{k^2} \) for large \( k \).
3. Recall the p-series test: \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges if \( p > 1 \). Here, since the denominator is \( k^2 \), which corresponds to \( p = 2 > 1 \), the series \( \sum \frac{1}{k^2} \) converges.
4. Use the Comparison Test by comparing \( \frac{7 + \sin k}{k^2} \) with \( \frac{8}{k^2} \) (since 8 is an upper bound for the numerator). Since \( \sum \frac{8}{k^2} \) converges, and \( 0 \leq \frac{7 + \sin k}{k^2} \leq \frac{8}{k^2} \), the original series converges by the Comparison Test.
5. Conclude that the series \( \sum_{k=1}^{\infty} \frac{7 + \sin k}{k^2} \) converges absolutely because it is bounded by a convergent p-series with \( p = 2 \).

Key Concepts

Convergence and Divergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. If the sum does not approach a finite value, the series diverges. Understanding this distinction is fundamental to analyzing series behavior.

Comparison Test for Series Convergence

The Comparison Test involves comparing a given series to a second series whose convergence behavior is known. If the terms of the given series are smaller than those of a convergent series, it also converges. This test is useful when terms are positive and can be bounded.

Behavior of the p-Series

A p-series has the form ∑ 1/k^p and converges if and only if p > 1. Since the given series has terms involving 1/k², recognizing this helps determine convergence by comparing to a known convergent p-series with p=2.