Sum Of An Infinite GP

The sum of n terms of a GP series is calculated using simple formulas. However, it is possible to calculate the sum of infinite terms of the GP. One of the properties of a GP series is that when there are infinite terms in a series and the common ratio is less than one, the sum of such a series is a finite value. In the following article, we will learn about this property of a geometric progression (GP) series. 

Sum of an infinite GP

The sum of n terms of a GP series can be calculated by simply following the summation formula,

S = a. (rn – 1) / (r – 1)

But this formula can not be used to calculate the sum of all the terms in the GP. 

In calculations, the sum of any series of infinite numbers always equals infinity; one of the properties of a geometric progression series is that the sum of infinite terms of a GP series can be a finite value, given that the common ratio of the series is less than one.

Let us understand this property of geometric progression series by the following example,

Consider the summation of the following series of numbers,

S = 1/3 + 1/6 + 1/12 + 1/24 +…. …(i)

The numbers of the series make a GP series.

The common ratio of the series is,

r = (1/6)/(1/3)

r = 1/2,

Let us multiply the equation (i) with 1/2 on both sides, we get,

S/2 = 1/6 + 1/12 + 1/24, …(ii)

Now, subtracting the (ii) from series (i),

S/2 = 1/3,

Or, S = 2/3.

Therefore, the sum of the infinite terms of the given series is 2/3.

Sum of an infinite GP when |r| < 1

Let us derive the formula for the sum of the infinite terms of a GP by understanding the properties of a geometric progression.

Consider a GP series with an infinite number of terms and the first term a, and common ratio r,

Also, consider that value of the constant ratio is less than one, |r| < 1,

The sum of infinite terms of such a GP is given by,

 S = a + ar + ar2 + ar3 + … …(i)

Multiplying both sides of the series with the common ratio r,

r.S = ar + ar2 + ar3 + ar4 +… …(ii)

Now subtracting the series (ii) from the series (i),

S(1 -r) = a,

Or, S = a/(1 – r).

Solved examples

Let us look at some of the questions based on the sum of infinite GP:

Sum of an Infinite GP: Questions

Example: Find the sum of all the terms of the following series,

1/4, 1/12, 1/36,…

Solution: Adding the terms of the given series for the infinite terms,

S = 1/4 + 1/12 + 1/36 +… …(i)

The given series makes a GP series,

The common ratio of the series r,

r = 1/3,

Multiplying the series (i) with 1/3 on both sides,

S/3 = 1/12 + 1/36 + 1/108 +… …(ii)

Subtracting the series,

2.S/3 = 1/4

Or, S = 3/8.

The sum of the infinite terms of the given series is 3/8.

Example: Find the sum of all the terms of the following GP series using the summation formula,

1/2, 1/8, 1/32,…

Solution: For the given GP series,

First term (a) = 1/2,

Common ratio (r) = 1/4,

Since the common ratio of the GP series is less than one, we can find the sum of the infinite terms by,

S = a/(1 -r)

S = (1/2) / (3/4),

S = 2/3.

The sum of the infinite terms of the given GP series is 2/3.

Example: Calculate the sum of all the terms of the following series,

1/10, 1/100, 1/1000,…

Solution: For the given series,

Adding all the terms of the series,

S = 1/100 + 1/1000 +1/10000 +… …(i)

The series (i) makes a GP series,

First term (a) = 1/100,

Common ratio (r) = 1/10,

Multiplying series (i) with 1/10,

S/10 = 1/1000 + 1/10000 + 1/100000 +… …(ii)

Subtracting the series,

9.S/10 = 1/100,

And, S = 1/90.

 The sum of the infinite numbers of the given series is 1/90.

Conclusion

It is impossible to calculate the sum of an infinite series; the sum of such series is considered infinity. But, it is one of the properties of geometric progression series that if the value of |r| is less than one, the sum of all the terms can be calculated in finite terms. We can also derive the calculation of the sum of infinite terms of a GP series.