Infinite Geometric Series

How to find the value of an infinite geometric series: formula, examples, and their solutions.

Formula

The infinite series, S, means the infinite sum:
S = a₁ + a₂ + a₃ + ....

For a geometric sequence
whose common ratio [r] is [-1 < r < 1],
the infinite geometric series, S, is
S = a₁/(1 - r).

Geometric series

Example 1

The first term is 1.
So a₁ = 1.

See the terms of the given series.
(1/2) / 1 = 1/2
(1/4) / (1/2) = 1/2
(1/8) / (1/4) = 1/2
...
So r = 1/2.

a₁ = 1
r = 1/2

So S = 1/(1 - 1/2).

1 - 1/2 = 1/2

Multiply 2
to both of the numerator and the denominator.

Then S = 2.

Example 2

See the given summation.
n goes from 1 to infinity.

So the given summation is an infinite series.

Sigma notation

a_n = 6⋅(1/7)ⁿ

So a₁ = 6⋅(1/7).

So a₁ = 6/7.

a_n = 6⋅(1/7)ⁿ

As n increases +1,
a_n changes ×(1/7).

So the given summation is
an infinite geometric series.

So r = 1/7.

Geometric sequence

a₁ = 6/7
r = 1/7

So S = (6/7) / (1 - 1/7).

1 - 1/7 = 6/7

The numerator and the denominator are the same:
6/7.

So S = 1.

Example 3

10 = 10
-20/3 = 10⋅(-2/3)
+40/9 = 10⋅(-2/3)²
-80/27 = 10⋅(-2/3)³
...

So a₁ = 10.
And r = -2/3.

a₁ = 10
r = -2/3

So S = 10 / [1 - (-2/3)].

1 - (-2/3) = 3/3 + 2/3

10 = 10/1

3/3 + 2/3 = 5/3

(10/1) / (5/3) = (10⋅3)/(1⋅5)
= 30/5

Complex fraction

30/5 = 6

So S = 6.

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