Geometric Series

Learning Outcomes

• Write the first n terms of a geometric sequence.
• Determine whether the sum of an inﬁnite geometric series exists.
• Give the sum of a convergent infinite geometric series.
• Solve an annuity problem using a geometric series.

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, $r$. We can write the sum of the first $n$ terms of a geometric series as

${S}_{n}={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+…+{a}_{1}{r}^{n - 1}$.

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

$r{S}_{n}={a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+…+{a}_{1}{r}^{n}$

Next, we subtract this equation from the original equation.

\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}

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 ${S}_{n}$, factor $a_1$ on the right hand side and divide both sides by $\left(1-r\right)$.

${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\text{ r}\ne \text{1}$

A General Note: Formula for the Sum of the First n Terms of a Geometric Series

A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first $n$ terms of a geometric sequence is represented as

${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\text{ r}\ne \text{1}$

How To: Given a geometric series, find the sum of the first n terms.

1. Identify ${a}_{1},r,\text{and}n$.
2. Substitute values for ${a}_{1},r$, and $n$ into the formula ${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}$.
3. Simplify to find ${S}_{n}$.

Example: Finding the First n Terms of a Geometric Series

Use the formula to find the indicated partial sum of each geometric series.

1. ${S}_{11}$ for the series $8 + -4 + 2 + \dots$
2. $\sum\limits _{k=1}^6 3\cdot {2}^{k}$

Try It

Use the formula to find the indicated partial sum of each geometric series.
${S}_{20}$ for the series $1\text{,}000 + 500 + 250 + \dots$

Use the formula to determine the sum $\sum\limits _{k=1}^{8}{3}^{k}$

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 $2+4+6+8+\dots$.

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

$1+0.2+0.04+0.008+0.0016+\dots$

The common ratio is $r=0.2$. As n gets large, the values of 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 $-1<r<1$.

DETERMINING WHETHER THE SUM OF AN INFINITE GEOMETRIC SERIES IS DEFINED

The sum of an infinite series is defined if the series is geometric and $-1<r<1$.

How To: Given the first several terms of an infinite series, determine if the sum of the series exists.

1. Find the ratio of the second term to the first term.
2. Find the ratio of the third term to the second term.
3. Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.
4. If a common ratio, r, was found in step 3, check to see if $-1<r<1$. If so, the sum is defined. If not, the sum is not defined.

Example: Determining Whether the Sum of an Infinite Series is Defined

Determine whether the sum of each infinite series is defined.

1. $12+8+4+\dots$
2. $\dfrac{3}{4}+\dfrac{1}{2}+\dfrac{1}{3}+\dots$
3. $\sum\limits _{k=1}^{\infty}{27}\cdot\left(\dfrac{1}{3}\right)^k$
4. $\sum\limits _{k=1}^{\infty}{5k}$

try it

Determine whether the sum of the infinite series is defined.

1. $\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{3}{4}+\dfrac{9}{8}+\cdots$
2. $24+(-12)+6+(-3)+\dots$
3. $\sum\limits _{k=1}^{\infty} 15\cdot(-0.3)^k$

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.

${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}$

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

\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}

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

\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}

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

A General Note: FORMULA FOR THE SUM OF AN INFINITE GEOMETRIC SERIES

The formula for the sum of an infinite geometric series with $-1<r<1$ is:

$S=\dfrac{{a}_{1}}{1-r}$

How To: Given an infinite geometric series, find its sum.

1. Identify $a_1$ and r.
2. Confirm that $-1<r<1$.
3. Substitute values for $a_1$ and r into the formula, $S=\dfrac{{a}_{1}}{1-r}$.
4. Simplify to find S.

Example: Finding the Sum of an Infinite Geometric Series

Find the sum, if it exists, for the following:

1. $10+9+8+7+\dots$
2. $248.6+99.44+39.776+\dots$
3. $\sum\limits _{k=1}^{\infty}4\text{,}374\cdot\left(-\dfrac{1}{3}\right)^{k-1}$
4. $\sum\limits _{k=1}^{\infty}\dfrac{1}{9}\cdot\left(\dfrac{4}{3}\right)^{k}$

Example: Finding an Equivalent Fraction for a Repeating Decimal

Find the equivalent fraction for the repeating decimal $0.\overline{3}$.

try it

Find the sum if it exists.

1. $2+\dfrac{2}{3}+\dfrac{2}{9}+\dots$
2. $\sum\limits _{k=1}^{\infty}{0.76k+1}$
3. $\sum\limits _{k=1}^{\infty}\left(-\dfrac{3}{8}\right)^k$

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 ${a}_{1}=50$ and $r=100.5\%=1.005$. 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 $n$ deposits using the formula for the sum of the first $n$ terms of a geometric series. In 6 years, there are 72 months, so $n=72$. We can substitute ${a}_{1}=50, r=1.005,$ and $n=72$ into the formula, and simplify to find the value of the annuity after 6 years.

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

Try It

At the beginning of each month, \$200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?

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