a.Does the sequence { k/(k + 1) } converge? Why or why not?

Verified step by step guidance

1. Identify the general term of the sequence, which is given by \(a_k = \frac{k}{k + 1}\).
2. Recall that to determine if a sequence converges, we need to find the limit of \(a_k\) as \(k\) approaches infinity, i.e., compute \(\lim_{k \to \infty} \frac{k}{k + 1}\).
3. To find this limit, divide the numerator and denominator by \(k\) to simplify the expression: \(\frac{k}{k + 1} = \frac{\frac{k}{k}}{\frac{k + 1}{k}} = \frac{1}{1 + \frac{1}{k}}\).
4. Evaluate the limit of the simplified expression as \(k\) approaches infinity: since \(\frac{1}{k} \to 0\), the expression approaches \(\frac{1}{1 + 0} = 1\).
5. Conclude that since the limit exists and equals 1, the sequence \(\left\{ \frac{k}{k + 1} \right\}\) converges, and its limit is 1.

Key Concepts

Definition of a Sequence

A sequence is an ordered list of numbers defined by a specific formula for its terms. Understanding how each term is generated, such as k/(k + 1), is essential to analyze its behavior as k increases.

Limit of a Sequence

The limit of a sequence is the value that the terms approach as the index k goes to infinity. If the terms get arbitrarily close to a fixed number, the sequence converges to that limit.

Convergence Criteria

A sequence converges if its limit exists and is finite. To determine convergence, evaluate the limit of the general term as k approaches infinity and check if it approaches a specific number.