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Sums of Geometric Series

Sum of infinite and finite geometric series and categorization as convergent or divergent.

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Geometric Series

An advanced factoring technique allows you to rewrite the sum of a finite geometric series in a compact formula. An infinite geometric series is more difficult because sometimes it sums to be a number and sometimes the sum keeps on growing to infinity. When does an infinite geometric series sum to be just a number and when does it sum to be infinity?

Geometric Series

A geometric series is a sum of numbers whose consecutive terms form a geometric sequence. Recall the advanced factoring technique for the difference of two squares and, more generally, two terms of any power (5 in this case).

\begin{align*}a^2-b^2&=(a-b)(a+b) \\ a^5-b^5&=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4) \\ a^n-b^n&=(a-b)(a^{n-1}+ \cdots +b^{n-1})\end{align*}a2b2a5b5anbn=(ab)(a+b)=(ab)(a4+a3b+a2b2+ab3+b4)=(ab)(an1++bn1)

If the first term is one then \begin{align*}a=1\end{align*}a=1. If you replace \begin{align*}b\end{align*}b with the letter \begin{align*}r\end{align*}r, you end up with:

\begin{align*}1-r^n=(1-r)(1+r+r^2+ \cdots r^{n-1})\end{align*}1rn=(1r)(1+r+r2+rn1)

You can divide both sides by \begin{align*}(1-r)\end{align*}(1r)  because \begin{align*}r \ne 1\end{align*}r1.

\begin{align*}1+r+r^2+\cdots r^{n-1}=\frac{1-r^n}{1-r} \end{align*}1+r+r2+rn1=1rn1r

The left side of this equation is a geometric series with starting term 1 and common ratio of \begin{align*}r\end{align*}r. Note that even though the ending exponent of \begin{align*}r\end{align*}r is \begin{align*}n-1\end{align*}n1, there are a total of \begin{align*}n\end{align*}n terms on the left. To make the starting term not one, just scale both sides of the equation by the first term you want, \begin{align*}a_1\end{align*}a1.

\begin{align*}a_1+a_1r+a_1r^2+\cdots a_1r^{n-1}=a_1 \left(\frac{1-r^n}{1-r}\right)\end{align*}a1+a1r+a1r2+a1rn1=a1(1rn1r)

This is the sum of a finite geometric series.

     

To sum an infinite geometric series, you should start by looking carefully at the previous formula for a finite geometric series. As the number of terms get infinitely large \begin{align*}(n\rightarrow \infty)\end{align*}(n) one of two things will happen.

\begin{align*}a_1 \left(\frac{1-r^n}{1-r}\right)\end{align*}a1(1rn1r)

Option 1: The term \begin{align*}r^n\end{align*}rn will go to infinity or negative infinity. This will happen when \begin{align*}\left |r \right \vert \ge 1\end{align*}|r|1. When this happens, the sum of the infinite geometric series does not go to a specific number and the series is said to be divergent.

Option 2: The term \begin{align*}r^n\end{align*}rn will go to zero. This will happen when \begin{align*}\left |r \right \vert < 1\end{align*}|r|<1. When this happens, the sum of the infinite geometric series goes to a certain number and the series is said to be convergent.

One way to think about these options is think about what happens when you take \begin{align*}0.9^{100}\end{align*}0.9100 and \begin{align*}1.1^{100}\end{align*}1.1100.

\begin{align*}0.9^{100} &\approx 0.00002656 \\ 1.1^{100} &\approx 13780\end{align*}0.91001.11000.0000265613780

As you can see, even numbers close to one either get very small quickly or very large quickly.

The formula for calculating the sum of an infinite geometric series that converges is:

\begin{align*}\sum \limits_{i=1}^{\infty} a_1 \cdot r^{i-1}=a_1 \left(\frac{1}{1-r}\right)\end{align*}i=1a1ri1=a1(11r)

Notice how this formula is the same as the finite version but with \begin{align*}r^n=0\end{align*}rn=0, just as you reasoned.

A partial sum of an infinite sum is the sum of all the terms up to a certain point. Considering partial sums can be useful when analyzing infinite sums.

    

Examples

Example 1

Earlier, you were asked when a infinite geometric series sum to just a number. An infinite geometric series converges if and only if \begin{align*}\left |r\right \vert <1\end{align*}|r|<1. Infinite arithmetic series never converge.

Example 2

Compute the sum of the following infinite geometric series two ways, without using the infinite summation formula and using the infinite summation formula.

\begin{align*}0.2+0.02+0.002+0.0002+\cdots \end{align*}0.2+0.02+0.002+0.0002+

Without using the summation formula:

You can tell just by looking at the sum that the infinite sum will be the repeating decimal \begin{align*}0.\overline{2} \end{align*}0.2¯¯¯. You may recognize this as the fraction \begin{align*}\frac{2}{9}\end{align*}29, but if you don’t, this is how you turn a repeating decimal into a fraction.

Let \begin{align*}x=0.\overline{2}\end{align*}x=0.2¯¯¯

Then \begin{align*}10x=2.\overline{2}\end{align*}10x=2.2¯¯¯

Subtract the two equations and solve for \begin{align*}x\end{align*}x.

\begin{align*}10x-x&=2.\overline{2}-0.\overline{2} \\ 9x&=2 \\ x&=\frac{2}{9}\end{align*}10xx9xx=2.2¯¯¯0.2¯¯¯=2=29

With using the summation formula:

The first term of the sequence is \begin{align*}a_1=0.2\end{align*}. The common ratio is 0.1. Since \begin{align*}\left |0.1 \right \vert <1\end{align*}, the series does converge.

\begin{align*}0.2 \left(\frac{1}{1-0.1}\right)=\frac{0.2}{0.9}=\frac{2}{9}\end{align*}

Example 3

Why does an infinite series with \begin{align*}r=1\end{align*} diverge?

If \begin{align*}r=1\end{align*} this means that the common ratio between the terms in the sequence is 1. This means that each number in the sequence is the same. When you add up an infinite number of any finite numbers (even fractions close to zero) you will always get infinity or negative infinity. The only exception is 0. This case is trivial because a geometric series with an initial value of 0 is simply the following series, which clearly sums to 0:

\begin{align*}0+0+0+0+\cdots\end{align*}

Example 4

What is the sum of the first 8 terms in the following geometric series?

\begin{align*}4+2+1+\frac{1}{2}+\cdots\end{align*}

The first term is 4 and the common ratio is \begin{align*}\frac{1}{2}\end{align*}.

\begin{align*}SUM=a_1\left(\frac{1-r^n}{1-r}\right)=4 \left(\frac{1- \left(\frac{1}{2}\right)^8}{1- \frac{1}{2}}\right)=4 \left(\frac{\frac{255}{256}}{\frac{1}{2}}\right)=\frac{255}{32}\end{align*}

Example 5

You put $100 in a bank account at the end of every year for 10 years. The account earns 6% interest. How much do you have total at the end of 10 years?

The first deposit gains 9 years of interest: \begin{align*}100 \cdot 1.06^9\end{align*}

The second deposit gains 8 years of interest: \begin{align*}100 \cdot 1.06^8\end{align*}. This pattern continues, creating a geometric series. The last term receives no interest at all.

\begin{align*}100 \cdot 1.06^9+100 \cdot 1.06^8+\cdots 100 \cdot 1.06+100\end{align*}

Note that normally geometric series are written in the opposite order so you can identify the starting term and the common ratio more easily.

\begin{align*}a_1=100,r=1.06\end{align*}

The sum of the 10 years of deposits is:

\begin{align*}a_1\left(\frac{1-r^n}{1-r}\right)=100 \left(\frac{1-1.06^{10}}{1-1.06}\right)\approx \$1318.08\end{align*}

Review

Find the sum of the first 15 terms for each geometric sequence below.

1. \begin{align*}5,10,20,\ldots\end{align*}

2. \begin{align*}2,8,32,\ldots\end{align*}

3. \begin{align*}5,\frac{5}{2},\frac{5}{4},\ldots\end{align*}

4. \begin{align*}12,4,\frac{4}{3},\ldots\end{align*}

5. \begin{align*}\frac{1}{3},1,3,\ldots\end{align*}

For each infinite geometric series, identify whether the series is convergent or divergent. If convergent, find the number where the sum converges.

6. \begin{align*}5+10+20+\cdots\end{align*}

7. \begin{align*}2+8+32+\cdots\end{align*}

8. \begin{align*}5+\frac{5}{2}+\frac{5}{4}+\cdots\end{align*}

9. \begin{align*}12+4+\frac{4}{3}+\cdots\end{align*}

10. \begin{align*}\frac{1}{3}+1+3+\cdots\end{align*}

11. \begin{align*}6+2+\frac{2}{3}+\cdots\end{align*}

12. You put $5000 in a bank account at the end of every year for 30 years. The account earns 2% interest. How much do you have total at the end of 30 years?

13. You put $300 in a bank account at the end of every year for 15 years. The account earns 4% interest. How much do you have total at the end of 10 years?

14. You put $10,000 in a bank account at the end of every year for 12 years. The account earns 3.5% interest. How much do you have total at the end of 12 years?

15. Why don’t infinite arithmetic series converge?

Review (Answer)

To see the Review answers, open this PDF file and look for section 12.5. 

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Vocabulary

converge

If a series has a limit, and the limit exists, the series converges.

convergent

If a series has a limit, and the limit exists, the series is convergent.

divergent

If a series does not have a limit, or the limit is infinity, then the series is divergent.

diverges

If a series does not have a limit, or the limit is infinity, then the series diverges.

geometric series

A geometric series is a geometric sequence written as an uncalculated sum of terms.

infinite geometric series

An infinite geometric series is the sum of the terms in a geometric sequence that has an infinite number of terms.

infinite series

An infinite series is the sum of the terms in a sequence that has an infinite number of terms.

partial sum

A partial sum is the sum of the first ''n'' terms in an infinite series, where ''n'' is some positive integer.

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