40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) k⁸ / (k¹¹ + 3)

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{k^{8}}{k^{11} + 3} \). We want to determine whether this series converges or diverges.

Simplify the general term for large \(k\) by comparing the dominant powers in numerator and denominator. For large \(k\), \( k^{11} + 3 \approx k^{11} \), so the term behaves like \( \frac{k^{8}}{k^{11}} = k^{-3} \).

Recognize that the series resembles a \(p\)-series of the form \( \sum k^{-p} \) with \( p = 3 \). Recall that a \(p\)-series converges if and only if \( p > 1 \).

Use the Limit Comparison Test with the series \( \sum \frac{1}{k^{3}} \) to rigorously confirm convergence. Compute the limit \( L = \lim_{k \to \infty} \frac{\frac{k^{8}}{k^{11} + 3}}{\frac{1}{k^{3}}} \).

Evaluate the limit \(L\). If \( 0 < L < \infty \), then both series either converge or diverge together. Since \( \sum \frac{1}{k^{3}} \) converges, conclude that the original series converges.

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Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to see if the series sums to a finite value.

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06:52

Convergence of an Infinite Series

Comparison Test and Limit Comparison Test

These tests compare the given series to a known benchmark series. The Comparison Test checks if terms are smaller or larger than a convergent or divergent series, while the Limit Comparison Test uses the limit of the ratio of terms to determine convergence by comparing growth rates.

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07:45

Limit Comparison Test

Asymptotic Behavior of Terms

Understanding how the terms behave for large indices (k → ∞) helps simplify the series. For example, in k⁸ / (k¹¹ + 3), the dominant term in the denominator is k¹¹, so the term behaves like k⁸/k¹¹ = 1/k³, which guides the choice of comparison series.

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