Examples of Convergent Sequences of Real Numbers

We will now look at some examples of convergent sequences of real numbers.

Example 1

Show that the sequence $\left \{ \frac{3n^2}{n^2 + 1} \right \}_{n=1}^{\infty}$ converges to $3$.

Let $\epsilon > 0$ be given. We have that:

(1)

\begin{align} \quad \biggr \lvert \frac{3n^2}{n^2 + 1} - 3 \biggr \rvert = \biggr \lvert \frac{3n^2 - 3(n^2 + 1)}{n^2 + 1} \biggr \rvert = \biggr \lvert \frac{-3}{n^2 + 1} \biggr \rvert = \frac{3}{n^2 + 1} < \frac{3}{n^2} \end{align}

We want $\frac{3}{n^2} < \epsilon$ which is equivalent to $\sqrt{\frac{3}{\epsilon}} < n$. So choose $N > \sqrt{\frac{3}{\epsilon}}$. Then if $n \geq N$ we have that $n > \sqrt{\frac{3}{\epsilon}} = \frac{\sqrt{3}}{\sqrt{\epsilon}}$, and so $\frac{1}{n} < \frac{\sqrt{\epsilon}}{\sqrt{3}}$, and $\frac{1}{n^2} < \frac{\epsilon}{3}$, so:

(2)

\begin{align} \quad \biggr \lvert \frac{3n^2}{n^2 + 1} - 3 \biggr \rvert < 3 \cdot \frac{1}{n^2} <3 \cdot \frac{\epsilon}{3} = \epsilon \end{align}

Therefore $\left \{ \frac{3n^2}{n^2 + 1} \right \}_{n=1}^{\infty}$ converges to $3$.

Example 2

Show that the sequence $\left \{ \frac{2}{n!} \right \}_{n=1}^{\infty}$ converges to $0$.

Let $\epsilon > 0$ be given. Then:

(3)

\begin{align} \quad \biggr \lvert \frac{2}{n!} - 0 \biggr \rvert = \biggr \lvert \frac{2}{1 \cdot 2 \cdot ... \cdot (n-1) \cdot n} \biggr \rvert \leq \frac{2}{n} \end{align}

So choose $N > \frac{\epsilon}{2}$. Then if $n \geq N$ we have that $n > \frac{2}{\epsilon}$, so $\frac{1}{n} < \frac{\epsilon}{2}$ and so:

(4)

\begin{align} \quad \biggr \lvert \frac{2}{n!} - 0 \biggr \rvert \leq \frac{2}{n} < 2 \cdot \frac{\epsilon}{2} = \epsilon \end{align}

Therefore $\left \{ \frac{2}{n!} \right \}_{n=1}^{\infty}$ converges to $0$.