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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
