Find the Sum of an Infinite Geometric Series?

Geometric progression can be defined as arrangement of Numbers such that Ratio of two successive numbers is constant. Here initial number is given as 'a' and common ratio is given as 'd'. Now we will see process of finding the sum of Infinite Geometric Series. Iinfinite Geometric Progression is given by: a₁ + a₁r + a₁r² + a₁r³ + ….. + a₁r^{n – 1}. (If -1 < r < 1). Then sum of the infinite geometric series formula is given by:

=> Sꝏ = a₁ / 1 – r (| r | < 1).

Now we will find the sum of an infinite geometric series with the help of example.

Let we have series 5 + 2.5 + 1.25 + 0.625 + 0.3125.....

Here the first term is given by a₁ = 5 (initial value) and common ratio is r = 0.5

So the value of common ratio lies between -1 and 1, the series will converge to some value. Let's see the sum of the first few terms.

=> a₁ = 5,

=> a₁ + a₁r = 5 + 2.5 = 7.5,

=> a₁ + a₁r + a₁r² = 5 + 2.5 + 1.25 = 8.75,

=> a₁ + a₁r + a₁r² + a₁r³ = 5 + 2.5 + 1.25 + 0.625 = 9.375, if we continue this process, we will get the following sums. (correct to 9 decimal places):

So the sum of 5 term = 9.84375,

Sum of 6 term = 9.921875,

Sum of 7 term = 9.9609375,

Sum of 8 term = 9.98046875,

Sum of 9 term = 9.990234375,

Sum of 10 term = 9.955117188,

Sum of 11 term = 9.997558594,

Sum of 12 term = 9.998779297,

Sum of 6 term = 9.999389648, continue this process, here we will see that the sum is not more than value of 10. Using formula we will get the same result. As we know that the formula is given by:

=> Sꝏ = a₁ / 1 – r,

=> Sꝏ = 5 / 1 – 0.5,

On further solving we get: