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….
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:
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