# Infinite Geometric Series - TeacherTube Infinite Geometric Series For r >1, the expressions go to infinity, so there is no limit. For r <-1, the expressions alternate between big positive and big negative

numbers, so there is no limit. For r =-1, the expressions alternate between -1 and 1, so there is no limit. What is an infinite series? An infinite series is a series of numbers

that never ends being summed. Example: 1 + 2 + 3 + 4 + 5 + . Strangely, sometimes infinite series have a finite sum (stops at a number). Other times infinite series sum to an infinitely large number (no sum).

Infinite series can either Converge have a finite sum Diverge keep growing to infinity (no sum) Infinite GEOMETRIC series

Have a common ratio between terms. Many infinite series are not geometric. We are just going to work with geometric ones. Example: Does this series have a sum?

IMPORTANT! First, we have to see if there even is a sum. We do this by finding r. If | r | < 1, If -1 < r < 1 ) there is a finite sum we CAN find. If | r | 1, the series sums to infinity (no sum).

Lets find r. We find r by dividing the second term by the first. In calculator:

(1 4) (1 2) enter. Absolute value smaller than 1? Has a sum! Now to find the sum The sum of an infinite series

Variables: S = sum r = common ratio between terms a1 = first term of series What did we get as a sum? _____

We found the sum of the infinite series Does this converge or diverge? You try: Find the sum (if it exists) of:

1 2 + 4 8 + .. Remember, fist find r We can express infinite geometric sums with Sigma Notation.

Evaluate: Classwork: Page 653: #6 9, #22 25