Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio, [latex]r[/latex]. We can write the sum of the first [latex]n[/latex] terms of a geometric series as

[latex]{S}_{n}={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+…+{a}_{1}{r}^{n - 1}[/latex].

Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first [latex]n[/latex] terms of a geometric series. We will begin by multiplying both sides of the equation by [latex]r[/latex].

[latex]r{S}_{n}={a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+…+{a}_{1}{r}^{n}[/latex]

Next, we subtract this equation from the original equation.

[latex]\begin{align}{S}_{n}&={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+…+{a}_{1}{r}^{n - 1} \\ -r{S}_{n}&=-\left({a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+…+{a}_{1}{r}^{n}\right) \\ \hline \left(1-r\right){S}_{n}&={a}_{1}-{a}_{1}{r}^{n}\end{align}[/latex]

Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for [latex]{S}_{n}[/latex], factor [latex]a_1[/latex] on the right hand side and divide both sides by [latex]\left(1-r\right)[/latex].

[latex]{S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\text{ r}\ne \text{1}[/latex]

Using the Formula for the Sum of an Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first n terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is [latex]2+4+6+8+\dots[/latex].

This series can also be written in summation notation as [latex] \sum\limits _{k=1}^{\infty} 2k[/latex], 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:

[latex]1+0.2+0.04+0.008+0.0016+\dots[/latex]

The common ratio is [latex]r=0.2[/latex]. As n gets large, the values of of [latex]r^n[/latex] 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 [latex]-1<r<1[/latex] approach 0; the sum of a geometric series is defined when [latex]-1<r<1[/latex].

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.

[latex]{S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}[/latex]

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

[latex]\begin{align} &{\left(\frac{1}{2}\right)}^{2} = \frac{1}{4} \\&{\left(\frac{1}{2}\right)}^{3} = \frac{1}{8} \\&{\left(\frac{1}{2}\right)}^{4} = \frac{1}{16} \end{align}[/latex]

The value of [latex]r^n[/latex] decreases rapidly. What happens for greater values of n?

[latex]\begin{align} &{\left(\frac{1}{2}\right)}^{10} = \frac{1}{1\text{,}024} \\&{\left(\frac{1}{2}\right)}^{20} = \frac{1}{1\text{,}048\text{,}576} \\&{\left(\frac{1}{2}\right)}^{30} = \frac{1}{1\text{,}073\text{,}741\text{,}824} \end{align}[/latex]

As n gets large, [latex]r^n[/latex] gets very small. We say that as n increases without bound, [latex]r^n[/latex] approaches 0. As [latex]r^n[/latex] approaches 0, [latex]1-r^n[/latex] approaches 1. When this happens the numerator approaches [latex]a_1[/latex]. This gives us the formula for the sum of an infinite geometric series.

Annuities

At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.

We can find the value of the annuity right after the last deposit by using a geometric series with [latex]{a}_{1}=50[/latex] and [latex]r=100.5\%=1.005[/latex]. After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.

We can find the value of the annuity after [latex]n[/latex] deposits using the formula for the sum of the first [latex]n[/latex] terms of a geometric series. In 6 years, there are 72 months, so [latex]n=72[/latex]. We can substitute [latex]{a}_{1}=50, r=1.005,[/latex] and [latex]n=72[/latex] into the formula, and simplify to find the value of the annuity after 6 years.

[latex]{S}_{72}=\dfrac{50\left(1-{1.005}^{72}\right)}{1 - 1.005}\approx 4\text{,}320.44[/latex]

After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of [latex]72\left(50\right) = $3,600[/latex]. This means that because of the annuity, the couple earned $720.44 interest in their college fund.