Chapter B, Problem 6AYU

To determine

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

Answer to Problem 6AYU

Solution:

The given sequence converges and the limit is $\frac{5}{2}$.

Explanation of Solution

Given:

$\left\{a_n\right\} = \left\{\frac{5n}{2n + 1}\right\}$

Calculation:

$\begin{array}{l}\left\{a_n\right\} = \left\{\frac{5n}{2n + 1}\right\}\\ \left\{\frac{5n}{2n + 1}\right\} = \frac{5n}{2n + 1} = \frac{5n}{2n\left(1 + \frac{1}{2n}\right)} = \frac{5}{2\left(1 + \frac{1}{2n}\right)}\end{array}$

Hence the sequence converges. Therefore we will find the limit

$\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\text{lim}}\frac{5}{2\left(1+\frac{1}{2n}\right)}=\frac{5}{2}$