b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?

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Identify the general term of the series: \(a_k = \frac{k}{k+1}\).

Examine the behavior of the terms \(a_k\) as \(k\) approaches infinity by finding \(\lim_{k \to \infty} a_k\).

Calculate the limit: \(\lim_{k \to \infty} \frac{k}{k+1} = \lim_{k \to \infty} \frac{k}{k(1 + \frac{1}{k})} = \lim_{k \to \infty} \frac{1}{1 + \frac{1}{k}}\).

Since \(\lim_{k \to \infty} a_k = 1 \neq 0\), recall the necessary condition for series convergence: if the terms do not approach zero, the series cannot converge.

Conclude that because the terms do not approach zero, the series \(\sum_{k=1}^\infty \frac{k}{k+1}\) diverges.

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