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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…. 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: Page 653: #6 – 9, #22 – 25