CK-12-Pre-Calculus Concepts

12.5. Geometric Series http://www.ck12.org

12.5 Geometric Series

Here you will sum infinite and finite geometric series and categorize geometric series as convergent or divergent.
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?

Watch This

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/62267

http://www.youtube.com/watch?v=mYg5gKlJjHc James Sousa: Geometric Series

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/62269

http://www.youtube.com/watch?v=RLZXFhvdlV8 James Sousa: Infinite Geometric Series

Guidance

Recall the advanced factoring technique for the difference of two squares and, more generally, two terms of any
power (5 in this case).

a^2 −b^2 = (a−b)(a+b)

a^5 −b^5 = (a−b)(a^4 +a^3 b+a^2 b^2 +ab^3 +b^4 )

an−bn= (a−b)(an−^1 +···+bn−^1 )

If the first term is one thena=1. If you replacebwith the letterr, you end up with:
1 −rn= ( 1 −r)( 1 +r+r^2 +···rn−^1 )
You can divide both sides by( 1 −r) becauser 6 =1.
1 +r+r^2 +···rn−^1 =^11 −−rrn