# Sum of Infinite Geometric Series

Previously we learned about the technique about the sum of geometric series, this time let’s tackle the sum of infinite geometric series. I have a secret to confess, series is my favorites topic in algebra especially telescoping series. I first heard about that series when I was 21. The time I joined a very active math group in facebook called Elite Math Circle. One of the applications of this series is to find the exact value of repeating decimals. I will also give example about that here. Let’s start!

Formula:

Given an infinite geometric sum expression $a+ar+ar^2+\dots$ where 0<r<1, the sum(Sn) can be expressed as follows:

$S_n=a+ar+ar^2+ \dots$

$S_n=\displaystyle\frac{a}{1-r}$

Sample Problem 1:

Find the sum of $2+1+\frac{1}{2}+\frac{1}{4}+\dots$

Solution:

Using the formula,

$S_n=\displaystyle\frac{a}{1-r}$

where r=1/2 and a=2

$S_{\infty}=\displaystyle\frac{2}{1-\frac{1}{2}}$

$S_{\infty}=4$

Sample Problem 2:

Ball rolled 16 cm in the first second, 12 cm in the second seconds, 9 cm on the third. Find the total distance travelled by the ball until it stops. Assume that the motion of the ball continues until it stops.

Solution:

This is another application of sum of infinite geometric series. If you think like a physicist, you might be thinking how much is the deceleration of the ball or the reason why the ball is decelerating. But set that aside for a while.

Using our formula,

$S_n=\displaystyle\frac{a}{1-r}$

where r=9/12=12/16=3/4, and a=16

$S_{\infty}=\displaystyle\frac{16}{1-\frac{3}{4}}$

$S_{\infty}=64 cm$

Sample Problem 3:

Find the exact value of 0.1212121212….

Solution:

There is an easy way to solve this problem. The process was discussed in detailed here.

But using the sum of geometric series, here is the process.

We can express the following in this form.

$0.12+0.0012+0.000012+\dots$ where r=0.01

Using the formula,

$S_n=\displaystyle\frac{a}{1-r}$

$S_n=\displaystyle\frac{0.12}{1-0.01}$

$S_n=\displaystyle\frac{0.12}{.99}=\displaystyle\frac{0.12}{.99}=\displaystyle\frac{4}{33}$

Sample Problem 4:

Find the exact value of 0.345454545…

Solution:

Rewrite the given as the sum of $0.3+0.045+0.00045+0.0000045+\dots$

From the term 0.045 onwards the series forms a geometric progression with r=0.01

Using the formula,

$S_n=\displaystyle\frac{a}{1-r}$

where a=0.045 and r=0.01

$S_n=0.3+\displaystyle\frac{0.045}{1-0.01}$

$S_n=\displaystyle\frac{19}{55}$

Sample Problem 5:

Find the sum of the series

$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{12}+\dots$

Solution:

Rearrange the series in this manner,

$(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots)+(\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+\dots)$

These are two infinite geometric series so we can use our formula.

$S_n=\displaystyle\frac{a}{1-r}+\displaystyle\frac{a}{1-r}$

$S_n=\displaystyle\frac{1}{1-\frac{1}{2}}+\displaystyle\frac{\frac{1}{3}}{1-\frac{1}{2}}$

$S_n=\displaystyle\frac{8}{3}$

Sample Problem 6:

A rubber ball is made to fall from a height of 50 feet and is observed to rebound 2/3 of the distance it falls. How far will the ball travel before coming to rest if the ball continues to fall in this manner?

Solution:

The movement of the ball must be vertical. Just think that the figure above is a projected view of the movement of the ball. We’ll be having a hard time to show the movement if the lines are overlapping. The red diagonal must be vertical so I projected another line on the left.

The total distance travelled by the ball must be the upward and downward path travelled by the ball.

Let Sd be total distance travelled by the ball downward and Su is the total distance travelled by the ball upward.

$S_n=\displaystyle\frac{a}{1-r}$

$S_d=\displaystyle\frac{50}{1-\frac{2}{3}}$

$S_d=150 ft$

$S_u=\displaystyle\frac{a}{1-r}$ ,

this time our a=50(2/3)=100/3.

$S_u=\displaystyle\frac{\frac{100}{3}}{1-\frac{2}{3}}$

$S_u=100 ft$

Total distance = Sd+Su = 150+100=250 ft.

### Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.

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