11.4 – Infinite Geometric Series. Sum of an Infinite Geometric Series.

11.4 – Infinite Geometric Series

Sum of an Infinite Geometric Series

Sum of an Infinite Geometric Series

• The sum S of an infinite geometric series with -1<r<1 is given by

S = a1

1 – r

Sum of an Infinite Geometric Series

• The sum S of an infinite geometric series with -1<r<1 is given by

S = a1

1 – r

Ex. 1 Find the sum of each infinite geometric series, if possible.

Sum of an Infinite Geometric Series

• The sum S of an infinite geometric series with -1<r<1 is given by

S = a1

1 – r

Ex. 1 Find the sum of each infinite geometric series, if possible.

a) ½ + ⅜ + …

Sum of an Infinite Geometric Series

• The sum S of an infinite geometric series with -1<r<1 is given by

S = a1

1 – r

Ex. 1 Find the sum of each infinite geometric series, if possible.

a) ½ + ⅜ + …

r =

Sum of an Infinite Geometric Series

• The sum S of an infinite geometric series with

-1<r<1 is given by

S = a1

1 – r

Ex. 1 Find the sum of each infinite geometric series, if possible.

a) ½ + ⅜ + …

r = ⅜

½

Sum of an Infinite Geometric Series

• The sum S of an infinite geometric series with

-1<r<1 is given by

S = a1

1 – r

Ex. 1 Find the sum of each infinite geometric series, if possible.

a) ½ + ⅜ + …

r = ⅜ = ¾

½

Sum of an Infinite Geometric Series

• The sum S of an infinite geometric series with

-1<r<1 is given by

S = a1

1 – r

Ex. 1 Find the sum of each infinite geometric series, if possible.

a) ½ + ⅜ + …

r = ⅜ = ¾, S = a1

½ 1 – r

Sum of an Infinite Geometric Series

• The sum S of an infinite geometric series with

-1<r<1 is given by

S = a1

1 – r

Ex. 1 Find the sum of each infinite geometric series, if possible.

a) ½ + ⅜ + …

r = ⅜ = ¾, S = a1

½ 1 – r

= ½

1 – ¾

Sum of an Infinite Geometric Series

• The sum S of an infinite geometric series with

-1<r<1 is given by

S = a1

1 – r

Ex. 1 Find the sum of each infinite geometric series, if possible.

a) ½ + ⅜ + …

r = ⅜ = ¾, S = a1

½ 1 – r

= ½ = ½

1 – ¾ ¼

Sum of an Infinite Geometric Series

• The sum S of an infinite geometric series with

-1<r<1 is given by

S = a1

1 – r

Ex. 1 Find the sum of each infinite geometric series, if possible.

a) ½ + ⅜ + …

r = ⅜ = ¾, S = a1

½ 1 – r

= ½ = ½ = 2 1 – ¾ ¼

b) 1 – 2 + 4 – 8 + …

b) 1 – 2 + 4 – 8 + …

r =

b) 1 – 2 + 4 – 8 + …

r = -2

1

b) 1 – 2 + 4 – 8 + …

r = -2 = -2

1

b) 1 – 2 + 4 – 8 + …

r = -2 = -2

1

Since r is not -1<r<1, finding the sum of the series is not possible.

b) 1 – 2 + 4 – 8 + …

r = -2 = -2

1

Since r is not -1<r<1, finding the sum of the series is not possible.

∞

Ex. 2 Evaluate ∑ 20(-¼)n – 1

n=1

b) 1 – 2 + 4 – 8 + …

r = -2 = -2

1

Since r is not -1<r<1, finding the sum of the series is not possible.

∞

Ex. 2 Evaluate ∑ 20(-¼)n – 1

n=1

an = a1rn – 1

b) 1 – 2 + 4 – 8 + …

r = -2 = -2

1

Since r is not -1<r<1, finding the sum of the series is not possible.

∞

Ex. 2 Evaluate ∑ 20(-¼)n – 1

n=1

an = a1rn – 1

a1 = 20

b) 1 – 2 + 4 – 8 + …

r = -2 = -2 1

Since r is not -1<r<1, finding the sum of the series is not possible.

∞

Ex. 2 Evaluate ∑ 20(-¼)n – 1

n=1

an = a1rn – 1

a1 = 20

r = -¼

b) 1 – 2 + 4 – 8 + …

r = -2 = -2 1

Since r is not -1<r<1, finding the sum of the series is not possible.

∞

Ex. 2 Evaluate ∑ 20(-¼)n – 1

n=1

an = a1rn – 1

a1 = 20

r = -¼S = a1

1 – r

b) 1 – 2 + 4 – 8 + …

r = -2 = -2 1

Since r is not -1<r<1, finding the sum of the series is not possible.

∞

Ex. 2 Evaluate ∑ 20(-¼)n – 1

n=1

an = a1rn – 1

a1 = 20

r = -¼S = a1 = 20

1 – r 1 – (-¼)

b) 1 – 2 + 4 – 8 + …

r = -2 = -2 1

Since r is not -1<r<1, finding the sum of the series is not possible.

∞

Ex. 2 Evaluate ∑ 20(-¼)n – 1

n=1

an = a1rn – 1

a1 = 20

r = -¼S = a1 = 20 = 20

1 – r 1 – (-¼) 5/4

b) 1 – 2 + 4 – 8 + …

r = -2 = -2 1

Since r is not -1<r<1, finding the sum of the series is not possible.

∞

Ex. 2 Evaluate ∑ 20(-¼)n – 1

n=1

an = a1rn – 1

a1 = 20

r = -¼S = a1 = 20 = 20 = 16

1 – r 1 – (-¼) 5/4

Ex. 3 Write the following repeating decimals as fractions.

Ex. 3 Write the following repeating decimals as fractions.

__

a) 0.39

Ex. 3 Write the following repeating decimals as fractions.

__

a) 0.39 = 39

99

Ex. 3 Write the following repeating decimals as fractions.

__

a) 0.39 = 39 = 13

99 33

Ex. 3 Write the following repeating decimals as fractions.

__

a) 0.39 = 39 = 13

99 33

___

b) 0.246

Ex. 3 Write the following repeating decimals as fractions.

__

a) 0.39 = 39 = 13

99 33

___

b) 0.246 = 246

999

Ex. 3 Write the following repeating decimals as fractions.

__

a) 0.39 = 39 = 13

99 33

___

b) 0.246 = 246 = 82

999 333