Limit Laws Example

Example

Determine whether the sequence

converges.

Complete Solution

Using the limit laws for infinite sequence, we would evaluate

Because our limit evaluates to a finite number, the sequence converges (and it converges to 1/2).

The above example is trivial, but demonstrates why we need the limit laws. They allow us to evaluate limits of more complicated sequences.

Explanation of Each Step

Step (1)

We applied the definition of convergence of a sequence. Recall that to determine if a sequence is convergent we evaluate

and if this limit exists, the sequence converges. If it doesn't the sequence is divergent.

Step (2)

To make the limit easier to evaluate, we divided both the numerator and denominator by . This is as commonly used trick when evaluating limits.

Step (3)

Steps (4, 5)

These two steps consisted of evaluating our limits and simple algebraic manipulation and should be straightforward to students who are familiar with limits of functions.