Sums of Infinite Series

Only geometrics where |r| < 1 have an infinite sum

S = t1/ 1 - r

To find where an infinite geometric converges set |r| < 1 and solve for x

To write a repeating decimal as a fraction what’s repeating/place – 1

Ex 1. Find the sum of the infinite geometric series

9 - 6 + 4 - … -6/9 = -2/3 4/-6 = -2/3 r = -2/3

S = 9/1 - (-2/3) = 9/(5/3)

S = 27/5

Ex 2. Find the ratio of a geometric with an infinite sum

24 and first term 12

S = 12/ 1 - r

12 = 24(r - 1) Divide by 24

1/2 = 1 - r

-1/2 = -r r = 1/2

Ex 3. For what values does the series converge

1 + (x - 2) + (x -2)^2 + (x - 3)^3 + …

(x - 2)/1 = x - 2

(x - 2)^2/(x - 2) = x - 2

|x - 2| < 1

-1 < x - 2 < 1 add 2 to both sides

1 < x < 3