Geometric Series

In a geometric sequence, the ratio between consecutive terms is constant. The ratio is called the common ratio. Ex. 5, 15, 45, 135,... 5*3=15 15*3=45 45*3=135 There is a common ratio of 3. This is a geometric sequence.

A geometric series is the expression for the sum of the terms of a geometric sequence. You can use a formula to evaluate a finite geometric series.

Sum of a Finite Geometric Series S n = a 1 (1-r n ) 1-r a 1 is the first term, r is the common ratio, and n is the number of terms

Use the formula to evaluate the series The common ratio is 2. The first term is 3. The number of terms is 6. S n = a 1 (1-r n ) 1-r = 3(1-2 6 ) 1-2 = 189

For each problem, identify a 1, r, and n for each series. Then evaluate each series. a b a 1 = -45 r = -3 n = 5 a 1 = 1/3 r = 1/3 n=4 S n = 2745 S n = -80/729

In some cases, you can evaluate an infinite geometric series. ·When |r|<1, the series converges (gets closer and closer to the sum S) ·When |r|≥1, the series diverges (the series approaches no limit or goes on forever)

Decide whether the infinite geometric series diverges or converges. a b. Σ 5(2) n-1 n=1 ∞ common ration r = -1/3 |-1/3|<1, the series converges and the series has a sum. 5, 10, 20, 40, 80,... common ratio r=2 |2|≥1, the series diverges and does not have a sum.

Sum of an Infinite Geometric Series S= a 1 1-r where a 1 is the first term and r is the common ratio

Evaluate each infinite geometric series. a b a 1 = 1 r = 1/2 a 1 = 3 r = 1/2 S= 1 = 2 1-1/2 S= 3 =6 1-1/2