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.

 

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