Question

Let \(a_{n}=\frac{n+1}{n}\). Discuss the convergence of \(\left\\{a_{n}\right\\}\) and \(\sum_{n=1}^{\infty} a_{n}\).

Short answer

The limit of the sequence \(a_{n}\) as \(n\) approaches infinity is 1, and the series \(\sum_{n=1}^{\infty} a_{n}\) diverges.

Step-by-step solution

Determine the Sequence Limit

In the first part, to find the limit of the sequence \(a_{n}\) as \(n\) approaches infinity, divide each term in the fraction by \(n\). The sequence becomes: \(a_{n}=1+\frac{1}{n}\). As \(n\) approaches infinity, \(\frac{1}{n}\) approaches 0. Therefore, the limit of the sequence \(a_{n}\) as \(n\) approaches infinity is 1.

Check the Infinity Series Convergence

To check the convergence of the series \(\sum_{n=1}^{\infty} a_{n}\), apply the nth Term Test. The nth Term Test states that if \(\lim_{{n \to \infty}} a_{n} ≠ 0\), then the series \(\sum_{n=1}^{\infty} a_{n}\) diverges. Since, from Step 1, \(\lim_{{n \to \infty}} a_{n} = 1 ≠ 0\), this implies that the series \(\sum_{n=1}^{\infty} a_{n}\) diverges.