Chapter B, Problem 21AYU

To determine

To find: Determine whether each sequence converges or diverges. If it converges, find its limit.

Answer to Problem 21AYU

Solution:

The given sequence converges. The limit is 1.

Explanation of Solution

Given:

$\left\{a_n\right\} = \left\{\frac{n + {\left(-1\right)}^n}{n}\right\}$

Calculation:

$\left\{a_n\right\} = \left\{\frac{n + {\left(-1\right)}^n}{n}\right\} = \frac{n}{n} + \frac{{\left(-1\right)}^n}{n} = 1 + \frac{{\left(-1\right)}^n}{n}$

Hence the sequence converges. Therefore we will find the limit

$\underset{n \to \infty }{\lim } a_n = \underset{n \to \infty }{\lim } \left\{\frac{n + {\left(-1\right)}^n}{n}\right\} = \underset{n \to \infty }{\lim } 1 + \frac{{\left(-1\right)}^n}{n} = 1$