1. Infinite Geometric Series

2. 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.

3. What is an infinite series?
An infinite series is a series of numbers that never ends being summed.
Example: ….
Strangely, sometimes infinite series have a finite sum (stops at a number).
Other times infinite series sum to an infinitely large number (no sum).

4. Infinite series can either…
Converge – have a finite sum
Diverge – keep growing to infinity (no sum)

5. Infinite GEOMETRIC series…
Have a common ratio between terms.
Many infinite series are not geometric. We are just going to work with geometric ones.

6. 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).
Let’s find r….

7. 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…

8. The sum of an infinite series…
Variables:
S = sum
r = common ratio between terms
a1 = first term of series

9. What did we get as a sum? _____
We found the sum of the infinite series
Does this converge or diverge?

10. You try: 1 – 2 + 4 – 8 + ….. Find the sum (if it exists) of:
Remember, fist find r…

11. We can express infinite geometric sums with Sigma Notation.

12. Evaluate:

13. Classwork:
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