Sequences

A sequence $\left\{\vphantom{ a_{n}}\right.$an$\left.\vphantom{ a_{n}}\right\}_{{n=1}}^{{\infty }}$ is a function whose domain is the set of positive integers. Calculate limits of sequences in Scientific Notebook by selecting an expression such as $\lim_{{n\rightarrow \infty }}^{}$$\left(\vphantom{ 1+% \frac{1}{n}}\right.$1 + ${\frac{{1}}{{n}}}$$\left.\vphantom{ 1+% \frac{1}{n}}\right)^{{n}}_{}$ and choosing Evaluate, or by defining an, writing $\lim_{{n\rightarrow \infty }}^{}$an, and choosing Evaluate.

To define an, with the insertion point in the equation an = $\left(\vphantom{ 1+\frac{1}{n}}\right.$1 + ${\frac{{1}}{{n}}}$$\left.\vphantom{ 1+\frac{1}{n}}\right)^{{n}}_{}$, choose New Definition from the Define submenu. You will get a dialog box that asks you how you wish to interpret the subscript. Choose A Function Argument.

$\blacktriangleright$ Evaluate

$\lim\limits_{{n\rightarrow \infty }}^{}$an = e

$\lim\limits_{{n\rightarrow \infty }}^{}$$\left(\vphantom{ 1+\frac{1}{n}}\right.$1 + ${\frac{{1}}{{n}}}$$\left.\vphantom{ 1+\frac{1}{n}}\right)^{{n}}_{}$ = e


A sequence such as $\left\{\vphantom{ \left( 1+\frac{1}{n}% \right) ^{n}}\right.$$\left(\vphantom{ 1+\frac{1}{n}% }\right.$1 + ${\frac{{1}}{{n}% }}$$\left.\vphantom{ 1+\frac{1}{n}% }\right)^{{n}}_{}$$\left.\vphantom{ \left( 1+\frac{1}{n}% \right) ^{n}}\right\}_{{n=1}}^{{\infty }}$ can be visualized graphically by plotting the expression $\left(\vphantom{ 1+\frac{1}{n}}\right.$1 + ${\frac{{1}}{{n}}}$$\left.\vphantom{ 1+\frac{1}{n}}\right)^{{n}}_{}$ at integer values of n. To get the following plot, set the Domain Interval to ≤n≤49, as Plot Style choose Point, and for Point Symbol choose Circle.

$\blacktriangleright$ Plot + Rectangular

$\left(\vphantom{ 1+\frac{1}{n}}\right.$1 + ${\frac{{1}}{{n}}}$$\left.\vphantom{ 1+\frac{1}{n}}\right)^{{n}}_{}$

dtbpF3in2.0003in0pt

You can generate this figure by plotting $\left(\vphantom{ 1+\frac{1}{n}}\right.$1 + ${\frac{{1}}{{n}}}$$\left.\vphantom{ 1+\frac{1}{n}}\right)^{{n}}_{}$, then revising the Plot Components page so that the Plot Style is Point, the Point Symbol is Circle, the Domain Interval is 1 to 49, and the Sample Size is 49.

This plot indicates that $\lim_{{n\rightarrow \infty }}^{}$$\left(\vphantom{ 1+\frac{1}{n}% }\right.$1 + ${\frac{{1}}{{n}% }}$$\left.\vphantom{ 1+\frac{1}{n}% }\right)^{{n}}_{}$ $\approx$ 2.7. Indeed, Evaluate yields e and Evaluate Numerically produces e = 2.718281828.