13.4 Series and their notations  (Page 4/18)

This series can also be written in summation notation as ${\displaystyle \sum _{k=1}^{\infty }2k,}$ where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges .

Determining whether the sum of an infinite geometric series is defined

If the terms of an infinite geometric series approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:

The common ratio $r\text{ = 0}\text{.2}.$ As $n$ gets very large, the values of $r^n$ get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with $-1<r<1$ approach 0; the sum of a geometric series is defined when $-1<r<1.$

Determine whether the sum of the infinite series is defined.

Finding sums of infinite series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first $n$ terms of a geometric series.

We will examine an infinite series with $r=\frac{1}{2}.$ What happens to $r^n$ as $n$ increases?

The value of $r^n$ decreases rapidly. What happens for greater values of $n?$