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:
The series converges because each term gets smaller and smaller (since -1 < r < 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:
Find the value of the infinite geometric series:
Didn't find what you are looking for on this page? Try search:
Online Algebra Solver
This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)
Go to: Online algebra solver
Ready for a break?
Play a math game.
(Well, not really a math game, but each game was made using math...)
The IntMath Newsletter
Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!
Short URL for this Page
Save typing! You can use this URL to reach this page:
Algebra Lessons on DVD
Easy to understand algebra lessons on DVD. See samples before you commit.
More info: Algebra videos