11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 0 to ∞)3k / ∜(k⁴ + 3)

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First, write down the general term of the series: \(a_k = \frac{3k}{\sqrt[4]{k^4 + 3}}\).

Next, analyze the behavior of the term \(a_k\) as \(k\) approaches infinity to understand if the terms approach zero, which is necessary for convergence.

Simplify the denominator for large \(k\): since \(k^4\) dominates \(3\), approximate \(\sqrt[4]{k^4 + 3} \approx \sqrt[4]{k^4} = k\).

Using this approximation, the term behaves like \(a_k \approx \frac{3k}{k} = 3\) for large \(k\), which does not approach zero.

Since the terms \(a_k\) do not approach zero, by the Test for Divergence (also called the nth-term test), the series \(\sum_{k=0}^\infty \frac{3k}{\sqrt[4]{k^4 + 3}}\) diverges.

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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 conclude convergence or divergence by relating to simpler series like p-series.

Asymptotic Behavior of Terms

Understanding how the terms behave for large indices is crucial. Simplifying expressions like 3k / ∜(k⁴ + 3) to dominant terms (e.g., 3k / k) helps identify the series' growth rate and guides the choice of appropriate convergence tests.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 0 to ∞)3k / ∜(k⁴ + 3)