Science:Infinite Series Module/Units/Unit 1/1.1 Infinite Sequences/1.1.04 Relationship to Limits of Functions

Our definition of convergence for an infinite sequence may look like the definition of a limit for functions.

Theorem: Relation to Limits of Functions
If we have that $\begin{array}{r}\underset{n\to \mathrm{\infty }}{\lim }f(x)=L\end{array}$ and f(n) = an, where x is a real number and n is an integer, then $\underset{n\to \mathrm{\infty }}{\lim }{a}_n=L.$

The above theorem makes it easier for us to evaluate limits of sequences.

Simple Example

Suppose we wish to determine whether the sequence

$\begin{array}{r}{a}_n=\frac{1}{n^p}\end{array}$

converges, where p is any positive integer.

Since we know that

$\begin{array}{r}\underset{x\to \mathrm{\infty }}{\lim }\frac{1}{x^p}=0,x\in \mathbb{ℝ}\end{array}$

for any integer p greater than zero, our theorem yields

$\begin{array}{r}\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n^p}=0,n\in {\mathbb{\mathbb{Z}}}^{+}.\end{array}$

Therefore, the given sequence

$\begin{array}{r}{a}_n=\frac{1}{n^p}\end{array}$

converges.