http://www.ck12.org Chapter 12. Discrete Math
The left side of this equation is a geometric series with starting term 1 and common ratio ofr. Note that even though
the ending exponent ofrisn−1, there are a total ofnterms on the left. To make the starting term not one, just scale
both sides of the equation by the first term you want,a 1.
a 1 +a 1 r+a 1 r^2 +···a 1 rn−^1 =a 1 (^11 −−rrn)
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(n→∞)one of two things will happen.
a 1 (^11 −−rrn)
Option 1:The termrnwill go to infinity or negative infinity. This will happen when|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 bedivergent.
Option 2:The termrnwill go to zero. This will happen when|r|<1. When this happens, the sum of the infinite
geometric series goes to a certain number and the series is said to beconvergent.
One way to think about these options is think about what happens when you take 0. 9100 and 1. 1100.
0. 9100 ≈ 0. 00002656
1. 1100 ≈ 13780
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:
∞
i=∑ 1 a^1 ·ri−^1 =a^1 (^11 −r)
Notice how this formula is the same as the finite version but withrn=0, just as you reasoned.
Example A
Compute the sum of the following infinite geometric series.
0. 2 + 0. 02 + 0. 002 + 0. 0002 +···
Solution:You can tell just by looking at the sum that the infinite sum will be the repeating decimal 0.2. You may
recognize this as the fraction^29 , but if you don’t, this is how you turn a repeating decimal into a fraction.
Letx= 0. 2
Then 10x= 2. 2
Subtract the two equations and solve forx.
10 x−x= 2. 2 − 0. 2
9 x= 2
x=^29Example B
Why does an infinite series withr=1 diverge?
Solution:Ifr=1 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: