Infinite Series Formula

Infinite series formula

The infinite series formula is used to calculate the summation of a series whose terms are of infinite numbers. Here, both arithmetic and geometric progressions are discussed. 

In geometric series (here all the terms have the same common multiplier) this formula is used to calculate the summation of the total series. 

The formula of summation is = a / (1 – r) 

Where, 

a = the first term of the series

r = the common ratio between two consecutive terms of the series, and r lies between -1 to 1.

If r is greater than 1, then the value of the total summation doesn’t exist. 

In case of arithmetic progression, 

The sum of an infinite arithmetic series is positive infinity when the common difference is greater than zero. 

The sum of an infinite arithmetic progression reaches negative infinity when the common difference is less than zero. 

So, the primary formula is, 

Total summation of an infinite series is = a / (1 – r) 

Where,

a = first term of the series

r = common ratio of the series

Solved examples

1) Let’s assume there is a series whose terms are, 1/4, 1/16, 1/64, 1/256 etc. and this series reaches infinity. Using the infinity series summation formula, find out the sum of this infinite series. 

Answer. So, here the first term of the series is 1/4.

So, a = 1/4

The common ratio is, 

1/16 / 1/4 = 1/4

So, r = 1/4

So, if r is less than 1, the formula of summation is, 

a / (1 – r) 

Applying the value, we get

= (1/4) (1 – 1/4) 

= 1/4 / 3/4

= 1/4 * 4/3

= 1/3

So, the sum of the infinite series 1/4 + 1/16 + 1/64 + 1/256 to infinity is 1/3.

2) Using the infinite series formula, find out the summation of the infinite series, 1/2 + 1/6 + 1/18 + 1/54 up to infinity. 

Answer. Here, the first term of the series is, 1/2

So, a = 1/2

The common ratio is, 

1/6 / 1/2

= 1/3

So, r = 1/3

According to the formula if r is less than 1, the summation of the series = 

a / (1 – r) 

Applying the value we get, 

(1/2) / (1 – 1/3) 

= 1/2 / 2/3

= 1/2 * 3/2

= 3/4

So, the summation of the infinite series 1/2 + 1/6 + 1/18 + 1/54 up to infinity is 3/4.