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

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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

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

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

Simple Example

Suppose we wish to determine whether the sequence

converges, where p is any positive integer.

Since we know that

for any integer p greater than zero, our theorem yields

Therefore, the given sequence

converges.