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 $\text{[math]}$ and f(n) = a_n, where x is a real number and n is an integer, then $\text{[math]}$

Theorem: Relation to Limits of Functions

If we have that

$\text{[math]}$

and f(n) = a_n, where x is a real number and n is an integer, then

$\text{[math]}$

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

Simple Example

Suppose we wish to determine whether the sequence

$\text{[math]}$

converges, where p is any positive integer.

Since we know that

$\text{[math]}$

for any integer p greater than zero, our theorem yields

$\text{[math]}$

Therefore, the given sequence

$\text{[math]}$

converges.