Introduction to Infinite Series

If we add an infinite sequence, we have an infinite series.

Infinite Sequence 

Infinite Series

which is called an infinite series (or just a series) and is denoted, for short, by the symbol

Definition Given a series

If the sequence  is convergent and  exists as a real number, then the series   is called convergent and we write

The number S is called the sum of the series. If the sequence  is divergent, then the series is called divergent. Its sequence of partial sums   has the terms

EXAMPLE 4: Suppose we know that the sum of the first n terms of the series    an is

Then the sum of the series is the limit of the sequence  :

In Example 4 we were given an expression for the sum of the first n terms, but it’s

usually not easy to find such an expression.

EXAMPLE 5: To Determine if an Infinite Series Converges, we could consider the sequence of partial sums.

The partial sums create a sequence

1,5, 14, 30, ...

Converge or Diverge?

EXAMPLE 6: To Determine if an Infinite Series Converges, we could consider the sequence of partial sums.

the partial sums create a sequence

Converge or Diverge?