# Infinite Series Formula

Infinite series is one of the important concepts in mathematics. It tells about the sum of a series of numbers that do not have limits. If the series contains infinite terms, it is called an infinite series, and the sum of the first n terms, Sn, is called a partial sum of the given infinite series. If the partial sum, i.e. the sum of the first n terms, Sn, given a limit as n tends to infinity, the limit is called the sum to infinity of the series, and the result is called the sum of infinite of series.

## Sum of Infinite Series Formula

The sum of infinite for an arithmetic series is undefined since the sum of terms leads to ±∞.

The sum to infinity for a geometric series is also undefined when |r| > 1.

If |r| < 1, the sum to infinity of a geometric series can be calculated.

Thus, the sum of infinite series is given by the formula:

$$\begin{array}{l}\large \mathbf{S_{\infty}=\frac{a}{1-r}}\end{array}$$

Or

$$\begin{array}{l}\large \lim_{n\rightarrow \infty}S_n = S_{\infty}=\frac{a}{1-r}\end{array}$$

### Solved Examples

Question 1: Evaluate:

$$\begin{array}{l}\sum_{0}^{\infty }(\frac{1}{2})^{n}\end{array}$$

Solution:

The sum of given series is,

$$\begin{array}{l}\sum_{0}^{\infty }(\frac{1}{2})^{n}\end{array}$$

So the series can be written as,

$$\begin{array}{l}\sum_{0}^{\infty }(\frac{1}{2})^{n} = (\frac{1}{2})^{0}+(\frac{1}{2})^{1}+(\frac{1}{2})^{2}+(\frac{1}{2})^{3}+…………..\end{array}$$

Here,

First term, a = 1

Common ratio, r = 1/2.
Infinite series formula is,

$$\begin{array}{l}S_{\infty} = \frac{a}{1-r}\end{array}$$

So,

$$\begin{array}{l}S_{\infty} = \frac{1}{1-\frac{1}{2}} = \frac{2}{2-1}=2\end{array}$$

Question 2:

Find the sum to infinity for the series 24 + 12 + 6 + …. (if exist).

Solution:

Given,

64 + 16 + 4 + ….

Here,

First term, a = 64

Common ratio = 16/64 = 1/4

r = 1/4 < 1, so we can find the sum to infinity.

$$\begin{array}{l}S_{\infty} = \frac{a}{1-r}\end{array}$$

So,

$$\begin{array}{l}S_{\infty} = \frac{64}{1-\frac{1}{4}} = \frac{64\times 4}{4-1}=\frac{256}{3}\end{array}$$

Therefore, the sum of infinite terms of the given series is 256/3.