42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)k⁴ / √(9k¹² + 2)

검증된 단계별 안내

First, write down the general term of the series: $a_k = \frac{k^4}{\sqrt{9k^{12} + 2}}$.

To analyze convergence, simplify the expression inside the square root for large $k$. Notice that \$9k^{12}$ dominates $2$, so approximate the denominator as $\sqrt{9k^{12}} = 3k^6$ for large $k$.

Rewrite the term using this approximation: $a_k \approx \frac{k^4}{3k^6} = \frac{1}{3k^2}$ for large $k$.

Recognize that the series behaves like $\sum \frac{1}{k^2}$ for large $k$, which is a p-series with $p=2$.

Since a p-series $\sum \frac{1}{k^p}$ converges if $p > 1$, conclude that the original series converges by the Comparison Test.

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주요 개념

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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 the given series to a second series with known convergence properties. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test is useful when terms resemble simpler series.

Asymptotic Behavior and Simplification of Terms

Analyzing the dominant terms in the numerator and denominator for large k helps simplify the general term of the series. This simplification reveals the term's growth rate, which is crucial for applying convergence tests effectively, such as comparing to p-series or geometric series.

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.∑ (from k = 1 to ∞)k⁴ / √(9k¹² + 2)