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Learn about infinite geometric series, their convergence or divergence, and how to find the sum using common ratio and first term. Explore examples and how to work with geometric series in mathematics.
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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…
Classwork: Page 653:#6 – 9, #22 – 25