Sequences

A sequence is a function with the domain being the natural numbers \(\mathbb{N} \).

A sequence can be written like this \(a_0,a_1, a_2, \ldots \) or like this:

\[(a_n)_{n=0}^\infty \]

The limit of a sequence is written like this:

\[\lim_{n\rightarrow \infty}a_n=A \]

Exercises

Use a slider to represent \(n\).

Set the width of the slider to 500. Then find the limits, if they exist, of following sequences:

1. \(\displaystyle{\lim_{n \rightarrow \infty}1^n}\)
2. \(\displaystyle{\lim_{n \rightarrow \infty}0.99^n}\)
3. \(\displaystyle{\lim_{n \rightarrow \infty}1.01^n}\)
4. \(\displaystyle{\lim_{n \rightarrow \infty}\left( 1+\frac{1}{n}\right)}\)
5. \(\displaystyle{\lim_{n \rightarrow \infty}\left( 1+\frac{1}{n}\right)^n}\)