Skip to main content
Search IntMath
Close

450+ Math Lessons written by Math Professors and Teachers

5 Million+ Students Helped Each Year

1200+ Articles Written by Math Educators and Enthusiasts

Simplifying and Teaching Math for Over 23 Years

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.

3. Infinite Geometric Series

by M. Bourne

If `-1 < r < 1`, then the infinite geometric series

a1 + a1r + a1r2 + a1r3 + ... + a1rn-1

converges to a particular value.

This value is given by:

`S_oo=(a_1)/(1-r)\ (|r|<1)`

The series converges because each term gets smaller and smaller (since -1 < r < 1).

Example 1

For the series:

`5 + 2.5 + 1.25 + 0.625 + 0.3125... `,

the first term is given by a1 = 5 and the common ratio is r = 0.5.

Since the common ratio has value between `-1` and `1`, we know the series will converge to some value.

Let's do the sum of the first few terms:

a1 = 5

a1 + a1r = 5 + 2.5 = 7.5

a1 + a1r + a1r2 = 5 + 2.5 + 1.25 = 8.75

a1 + a1r + a1r2 + a1r3 = 5 + 2.5 + 1.25 + 0.625 = 9.375

Continuing this pattern, we will get the following sums (correct to 9 decimal places):

Sum to 5 terms `= 9. 6875`

Sum to 6 terms `= 9. 84375`

Sum to 7 terms `= 9. 921875`

Sum to 8 terms `= 9. 9609375`

Sum to 9 terms `= 9. 98046875`

Sum to 10 terms `= 9. 990234375`

Sum to 11 terms `= 9. 995117188`

Sum to 12 terms `= 9. 997558594`

Sum to 13 terms `= 9. 998779297`

Sum to 14 terms `= 9. 999389648`

Where do we use this?

See in a later chapter how we use the sum of an infinite GP and differentiation to find polynomial approximations for functions.

We also see how a calculator works, using these progressions.

We could keep going and would see that the sum gets closer and closer to, but does not go over `10`. It appears the infinite sum is 10.

Applying the formula now, we confirm our result:

`S_oo=(a_1)/(1-r)=5/(1-0.5)=5/0.5=10`

Example 2

Find the value of the infinite geometric series:

`4+2+1+1/2+1/4+1/8+...`

Answer

Here,

`a_1=4,\ r=1/2`

`s_oo=(a_1)/(1-r)`

`S_oo=4/(1-1/2)=4/(1/2)=8`

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.