infinite geometric series. for r >1, the expressions go to infinity, so there is no limit. for r
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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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