Infinite Geometric Series

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Presentation transcript:

Infinite Geometric Series

Sum of an infinite Geometric Series The sum S of an infinite geometric series where -1 < r < 1 is given by a1 1-r S = An infinite geometric series where r ≥ 1 does not have a sum.

Example 1. Find the sum of the infinite geometric series if it exists. 2 3 4 9 8 27 + + + ... First find the value of r to determine if the sum exists. a1 = 2 3 a2 = 4 9

Example 1. Find the sum of the infinite geometric series if it exists. a1 = 2 3 2 3 4 9 8 27 + + + ... a2 = 4 9 r = (4/9)/(2/3) r = (2/3) Sn = a1 1-r = 2/3 1-(2/3) = 2

Example 2. Find the sum of the infinite geometric series if it exists. 1 - 3 + 9 - 27 a1 = 1 a2 = -3 r = -3/1 = -3 The sum does not exist.

Example 3. You pull a pendulum back and release it. It follows a swing pattern of 25cm, 20cm, 16cm and so on until it comes to a rest. What is the total distance the pendulum swings before it comes to rest.

Example 3. Pendulum swings 25cm, 20cm,16cm,etc.Total distance travel The swing pattern of the pendulum forms the infinite geometric series 25, 20, 16, ... a1 = 25 a2 = 20 r = 20/25 = 4/5 Therefore the sum exists.

Example 3. Pendulum swings 25cm, 20cm,16cm,etc.Total distance travel a1 = 25 a2 = 20 r = 20/25 = 4/5 Sn = a1 1-r = 25 1-(4/5) = 125

The sum of an infinite geometric series can be used to express a repeating decimal in the form a/b. Repeating decimals such as 0.2 and 0.47 represent 0.22222... and 0.474747... respectively. Each expression can be written as an infinite geometric series.

Repeating decimals such as 0.2 and 0.47 represent 0.22222... and 0.474747... respectively. Each expression can be written as an infinite geometric series. 0.222... = 0.2+0.02+0.002+... 0.4747... = 0.47+0.0047+...

Example 4. Express 0.12 as a rational number of the form a/b. Rewrite as a geometric sum 0.12 = 0.12+0.0012+0.000012+... a1 = 0.12 a2 = 0.0012 r = 0.0012 0.12 r =0.01

Example 4.Express 0.12 as rational Sn = a1 1-r = 0.12 1-0.01 = 0.12 0.99 = 12 99 = 4 33

Sigma notation uses the general form of the nth term of a geometric Example 5. Evaluate ∑ 35(-1/4)n-1  n=1 Sigma notation uses the general form of the nth term of a geometric series, or a1rn-1. a1 = 35 r = -1/4

Example 5. Evaluate ∑ 35(-1/4)n-1 a1 = 35 r = -1/4 35 Sn = 1-(-1/4)  n=1 a1 = 35 r = -1/4 Sn = 35 1-(-1/4) = 28