Question

\({\bf{37 - 40}}\) Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

\({a_n} = n + \frac{1}{n}\)

Short answer

The sequence is increasing and does not have the upper bound. It is bounded below by\(2\).

Step-by-step solution

Definition

A sequence\(\left\{ {{a_n}} \right\}\)is called increasing if \({a_n} < {a_{n + 1}}\)for all, that is, \({a_1} < {a_2} < {a_3} < \cdots \). It is called decreasing if \({a_n} > {a_{n + 1}}\) for all. A sequence is monotonic if it is either increasing or decreasing.

For increasing or decreasing

To determine whether the sequence is increasing or decreasing calculate the difference between\({a_n}\)and\({a_{n + 1}}\).

\(\begin{aligned}{a_{n + 1}} - {a_n} &= \frac{{{{(n + 1)}^2} + 1}}{{n + 1}} - \frac{{{n^2} + 1}}{n}\\ &= \frac{{n\left( {{{(n + 1)}^2} + 1} \right) - (n + 1)\left( {{n^2} + 1} \right)}}{{n(n + 1)}}\\ &= \frac{{n\left( {{n^2} + 2n + 2} \right) - \left( {{n^3} + n + {n^2} + 1} \right)}}{{n(n + 1)}}\\ &= \frac{{{n^3} + 2{n^2} + 2n - {n^3} - n - {n^2} - 1}}{{n(n + 1)}}\\ &= \frac{{{n^2} + n - 1}}{{n(n + 1)}} > 0\;\;,\forall n \in N\end{aligned}\)

Thus, \({a_{n + 1}} - {a_n} > 0\)\( \Rightarrow {a_{n + 1}} > {a_n}\)

So the given sequence is increasing sequence.

For bound

Consider the bound.

For\(n \ge 1,{n^2} + 1 > 0\).

So, \(\frac{{{n^2} + 1}}{n} \ge 0\)as the ratio of positive numbers

For\(n = 1\), simplify as follows:

\(\begin{aligned}\frac{{{n^2} + 1}}{n} &= \frac{{{{(1)}^2} + 1}}{1}\\ &= 2\end{aligned}\)

So, the entire sequence \({a_n}\)is bounded below by 2.