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). Let’s find r….

In calculator:

(1 ÷ 4) ÷ (1 ÷ 2) enter.

Absolute value smaller than 1?

Has a sum! Now to find the sum…

We find r by dividing the second term by the first.

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:

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