## Pages

### Sum of an infinite Geometric Progression

This post is about the sum of an infinite Geometric Progression. For the sum of a finite Geometric Progression, read this post.

It seems impossible that we can find the sum of an infinite series of numbers. Since we do not know the value of infinity, nor the value of the term at infinity in a particular infinite Geometric Progression, it is not possible to get the exact sum of the infinite G.P., but it is possible to obtain a fairly accurate result. Here we will discuss the derivation to the formula that enables us to find the sum of an infinite Geometric Progression.

Let 'a' be the first term and 'r' be the common ratio of a Geometric Progression. We know that the sum of a finite number of terms, say 'n' terms, is
Sn = a(1 - r^n)/(1 - r)
For an infinite G.P., the value of 'n' in the above formula will be infinity. Now let 'r' be between -1 and 1 (i.e., |r| < 1). Then, r^n will become smaller and smaller as the value of 'n' increases. For example, if r = 0.5, then r^2 = 0.25, r^3 = 0.125 and r^10 = 0.0009 and so on.

Thus as n nears infinity, the value of r^n nears zero. Thus for n = infinity, we can take r^n = 0. Hence we get
S = a(1 - 0)/(1 - r)
That is,
S = a/(1 - r)
Thus the sum of an infinite number of terms of a Geometric Progression is obtained by the above formula when a is the first term and r the common difference.

Note that it is not necessary for a G.P. to have |r| < 1.  The common ratio can be greater than 1, but in that case finding the sum of an infinite G.P. will be different from that described above.

Related posts: