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 ${\begin{aligned}\lim _{n\rightarrow \infty }f(x)=L\end{aligned}}$ and f(n) = an, where x is a real number and n is an integer, then $\lim _{n\rightarrow \infty }a_{n}=L.$

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

Simple Example

Suppose we wish to determine whether the sequence

${\begin{aligned}a_{n}={\frac {1}{n^{p}}}\end{aligned}}$

converges, where p is any positive integer.

Since we know that

${\begin{aligned}\lim _{x\rightarrow \infty }{\frac {1}{x^{p}}}=0,\quad x\in \mathbb {R} \end{aligned}}$

for any integer p greater than zero, our theorem yields

${\begin{aligned}\lim _{n\rightarrow \infty }{\frac {1}{n^{p}}}=0,\quad n\in \mathbb {Z} ^{+}.\end{aligned}}$

Therefore, the given sequence

${\begin{aligned}a_{n}={\frac {1}{n^{p}}}\end{aligned}}$

converges.