Activity 8.3.7.
Consider the series \(\sum \frac{k+1}{k^3+2}\text{.}\) Since the convergence or divergence of a series only depends on the behavior of the series for large values of \(k\text{,}\) we might examine the terms of this series more closely as \(k\) gets large.
(a)
By computing the value of \(\frac{k+1}{k^3+2}\) for \(k = 100\) and \(k = 1000\text{,}\) explain why the terms \(\frac{k+1}{k^3+2}\) are essentially \(\frac{k}{k^3}\) when \(k\) is large.
(b)
Let’s formalize our observations in (a) a bit more. Let \(a_k = \frac{k+1}{k^3+2}\) and \(b_k = \frac{k}{k^3}\text{.}\) Calculate
\begin{equation*}
\lim_{k \to \infty} \frac{a_k}{b_k}\text{.}
\end{equation*}
What does the value of the limit tell you about \(a_k\) and \(b_k\) for large values of \(k\text{?}\) Compare your response from part (a).
(c)
Does the series \(\sum \frac{k}{k^3}\) converge or diverge? Why? What do you think that tells us about the convergence or divergence of the series \(\sum \frac{k+1}{k^3+2}\text{?}\) Explain.
