# Infinite Series Formula

The infinite series formula is used to find the sum of a sequence where the number of terms is infinite. There are various types of infinite series. In this section, we will discuss the sum of infinite arithmetic series and the sum of infinite geometric series. The arithmetic series is the sequence where the difference between each consecutive term is constant throughout and the geometric series is the series where the ratio of the consecutive terms to the preceding term is the same throughout. The infinite series formula is a handy tool to calculate the sum very quickly. Let us learn more about the infinite series formula along with solved examples.

## What Is Infinite Series Formula?

The sum of the infinite geometric series formula is used to find the sum of the series that extends up to infinity. This is also known as the sum of infinite GP. While finding the sum of a GP, we find that the sum converges to a value, though the series has infinite terms. The infinite series formula if −1<r<1, can be given as,

- Sum = a/(1-r)

Where,

- a = first term of the series
- r = common ratio between two consecutive terms and −1 < r < 1

Note: If r > 1, the sum does not exist as the sum does not converge.

- The sum of an infinite arithmetic sequence is ∞, if d > 0, or
- The sum of an infinite arithmetic sequence is ∞, if d > 0- ∞, if d < 0.

Let us now have a look at a few solved examples using the Infinite Series Formula.

## Examples using Infinite Series Formula

**Example 1:** **Using an infinite series formula, find the sum of infinite series: 1/4 + 1/16 + 1/64 + 1/256 +⋯**

**Solution:**

Given: a = ¼

r = (1/16) / (1/4) = (1/64) / (1/16) = ¼

To find: Sum of the given infinite series

If r<1 is then sum is given as Sum = a/(1-r)

Applying the values to the infinite series formula, we get

Sum=(1⁄4)/(1-1⁄4)

Sum=(1⁄4)/(3⁄4)

Sum=4/(3*4)

Sum=1/3

**Answer: The sum of 1/4+1/16+1/64+1/256+⋯ is 1/3**

**Example 2: Using the infinite series formula, find the sum of infinite series: 1/2 + 1/6 + 1/18 + 1/54 + ⋯**

**Solution:**

Given: a = 1/2

r = (1/6) / (1/2) = (1/18) / (1/6) = 1/3

To find: Sum of the given infinite series

If r<1 is then sum is given as Sum = a/(1-r)

Applying the values to the infinite series formula, we get

Sum=(1⁄2)/(1-1⁄3)

Sum=(1⁄2)/(2⁄3)

Sum=3/(2*2)

Sum=3/4

**Answer: The sum of 1/2 + 1/6 + 1/18 + 1/54 + ⋯ is 3/4**

**Example 3: Evaluate 3 + 7 + 11 + .......
Solution:**

a = 3, d = 4 and n = ∞

Here the difference > 0.

So, the sum = + ∞

**Answer: 3 + 7 + 11 + ....... ** **= + ∞**

## FAQs on Infinite Series Formula

### What Is the Sum of Infinite Terms?

An infinite series has an infinite number of terms. The sum of the first n terms, S_{n}, is called a partial sum. If S_{n} tends to a limit as n tends to infinity, the limit is called the sum to infinity of the series. The sum of infinite arithmetic series is either +∞ or - ∞. The sum of the infinite geometric series when the common ratio is <1, then the sum converges to a/(1-r), which is the infinite series formula of an infinite GP. Here a is the first term and r is the common ratio.

### What Is the Infinite Series Formula?

The sum of infinite arithmetic series is either +∞ or - ∞. The sum of the infinite geometric series formula is also known as the sum of infinite GP. The infinite series formula if the value of r is such that −1<r<1, can be given as,

Sum = a/(1-r)

Where,

- a = first term of the series
- r = common ratio between two consecutive terms and −1<r<1

### What Is a and r in Infinite Series Formula?

In finding the sum of the given infinite geometric series If r<1 is then sum is given as Sum = a/(1-r). In this infinite series formula, a = first term of the series and r = common ratio between two consecutive terms and −1<r<1.

### Find the sum of the infinite GP 0.3+ 0.33+ 0.333+....

This infinite GP can be written as, 3/10 + 3/100 + 3/1000+..........

Here we find that the first term a = 3/10, and r = 3/100 ÷ 3/10 = 1/ 10

Since r < 1, the sum must converge to a/ (1-r) as per the infinite series formula for infinite GP.

Thus the sum to infinity = (3/10) ÷ (1 - 1/10)

Sum = 3/10 ÷ 9/10 = 1/3

Thus the sum converges to 1/3. 0.3+ 0.33+ 0.333+.... = 1/3

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