What is the sum of the infinite geometric series

Before knowing the geometric series formula, let us recall what is a geometric series. It is a series (sum of terms) where the ratios of every two consecutive terms are the same. The geometric series formula includes:

• Formula to find the nth term of a geometric series.
• Finding the sum of a finite geometric series.
• Finding the sum of an infinite geometric series.

What is a Geometric Series?

A geometric series is the sum of finite or infinite terms of a geometric sequence. For the geometric sequence a, ar, ar2, ..., arn-1, ..., the corresponding geometric series is a + ar + ar2 + ..., arn-1 + .... We know that "series" means "sum". In particular, the geometric series means the sum of the terms that have a common ratio between every adjacent two of them. There can be two types of geometric series: finite and infinite. Here are some examples of geometric series.

• 1/2 + 1/4 + .... + 1/8192 is a finite geometric series
  where the first tern, a = 1/2 and the common ratio, r = 1/2
• -4 + 2 - 1 + 1/2 - 1/4 + ... is an infinite geometric series
  where the first term, a = -4 and the common ratio r = -1/2

Geometric Series Formula

The geometric series formula refers to the formula that gives the sum of a finite geometric sequence, the sum of an infinite geometric series, and the nth term of a geometric sequence. The sequence is of the form {a, ar, ar2, ar3, …….} where, a is the first term, and r is the "common ratio".

Geometric Series Formulas

The formulas for a geometric series include the formulas to find the nth term, the sum of n terms, and the sum of infinite terms. Let us consider a geometric series whose first term is a and common ratio is r.

a + ar + ar2 + ar3 + ...

Formula 1: The nth term of a geometric sequence is,

nth term = a rn-1

Where,

• a is the first term
• r is the common ratio of every two successive terms
• n is the number of terms.

To see how this formula is derived, click here.

Formula 2: The sum formula of a finite geometric series a + ar + ar2 + ar3 + ... + a rn-1 is

Sum of n terms = a (1 - rn) / (1 - r) (or) a (rn - 1) / (r - 1)

where,

• a is the first term
• r is the common ratio every two consecutive terms
• n is the number of terms.

To see how this formula is derived, click here.

Formula 3: The sum formula of an infinite geometric series a + ar + ar2 + ar3 + ... is

Sum of infinite geometric series = a / (1 - r)

where,

• a is the first term
• r is the common ratio every two successive terms

To see how this formula is derived, click here.

Convergence of Geometric Series

A finite geometric series always converges. But the convergence of an infinite geometric series depends upon the value of its common ratio. An infinite geometric series a, ar, ar2, ...

• converges when |r| < 1 and hence we can find its sum using the formula a / (1 - r).
• diverges when |r| > 1 and hence we can't find its sum in this case.

Examples:

The geometric series 2 + 4 + 8 + 16 + .... diverges as |r| = |2| = 2 > 1. So its sum can't be found.

The geometric series 1 - 1/2 + 1/4 - 1/8 + 1/16 - .... converges as |r| = |-1/2| = 1/2 < 1. Hence, its sum is, a / (1 - r) = 1 / (1 - (-1/2)) = 1/ (3/2) = 2/3.

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1. Example 1: Find the 10th term of the geometric series 1 + 4 + 16 + 64 + ...

   Solution:

   To find: The 10th term of the given geometric series.

   In the given series,

   The first term, a = 1.

   The common ratio, r = 4 / 1 (or) 16 / 4 (or) 64 / 16 = 4.

   Using the formulas of a geometric series, the nth term is found using:

   nth term = a rn-1

   Substitute n = 10, a = 1, and r = 4 in the above formula:

   10th term = 1 × 410-1 = 49 = 262,144

   Answer: The 10th term of the given geometric sequence = 262,144.

2. Example 2: Find the sum of the following geometric series: i) 1 + (1/3) + (1/9) + ... + (1/2187) ii) 1 + (1/3) + (1/9) + ... using the formula for a geometric series.

   Solution:

   To find: The sum of the given two geometric series.

   In both of the given series,

   the first term, a = 1.

   The common ratio, r = 1 / 3.

   i) In the given series,

   nth term = 1 / 2187

   a rn-1 = 1 / 37

   1 × ( 1 / 3) n-1 = 3-7

   3-n + 1 = 3-7

   -n + 1 = -7

   -n = -8

   n = 8

   So we need to find the sum of the first 8 terms of the given series.

   Using the sum of the finite geometric series formula:

   Sum of n terms = a (1 - rn) / (1 - r)

   Sum of 8 terms = 1 ( 1 - (1/3)8 ) / (1 - 1/3)

   = (1 - (1 / 6561)) / (2 / 3)

   = (6560 / 6561) × (3 / 2)

   = 3280 / 2187

   ii) The given series is an infinite geometric series.

   Using the sum of the infinite geometric series formula:

   Sum of infinite geometric series = a / (1 - r)

   Sum of the given infinite geometric series

   = 1 / (1 - (1/3))

   = 1 / (2 / 3)

   = 3 / 2

   Answer: i) Sum = 3280 / 2187 and ii) Sum = 3 / 2

3. Example 3: Calculate the sum of the finite geometric series if a = 5, r = 1.5 and n = 10.

   Solution:

   To find: the sum of geometric series

   Given: a = 5, r = 1.5, n = 10

   sn = a(1−rn)/(1−r)

   The sum of ten terms is given by S10 = 5(1−(1.5)10)/(1−1.5)

   = 566.65

   Answer: The sum of the geometric series is 566.65.

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A geometric sequence is the collection of terms where the ratio of every two successive terms is the same whereas a geometric series is the "sum" of the terms of the geometric sequence.

What Are the Geometric Series Formulas in Math?

The geometric series formulas are the formulas that help to calculate the sum of a finite geometric sequence, the sum of an infinite geometric series, and the nth term of a geometric sequence. These formulas are geometric series with first term 'a' and common ratio 'r' given as,

• nth term = a rn-1
• Sum of n terms = a (1 - rn) / (1 - r)
• Sum of infinite geometric series = a / (1 - r)

What is the Infinite Geometric Series Formula?

The sum formula of an infinite geometric series a + ar + ar2 + ar3 + ... can be calculated using the formula, Sum of infinite geometric series = a / (1 - r), where a is the first term, r is the common ratio for all the terms and n is the number of terms.

Is it possible to Find the Sum of the Geometric Series Always?

If the geometric series is finite, we can find its sum always. But if it is infinite, then its sum can be found only when the absolute value of its common ratio is less than 1.

When Does a Geometric Sequence Converge?

We know that 'r' represents the common ratio between every two consecutive terms of a geometric series. An infinite geometric series converges only when |r| < 1 and we can find its sum only if it converges.

How To Use the Geometric Series Formula?

To use the geometric series formula

• Step 1: Check for the given values, a, r and n.
• Step 2: Put the values in the geometric series formula as per the requirement - the sum of a finite geometric sequence, the sum of an infinite geometric series, or the nth term of a geometric sequence

What are the Applications of Geometric Series?

Geometric series formulas are used throughout mathematics. These have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.

What is 'r' in the Geometric Series Formula?

In the geometric series formula, 'r' refers to the common ratio. The formulas for geometric series with 'n' terms and the first term 'a' are given as,

Before learning the infinite geometric series formula, let us recall what is a geometric series. A geometric series is the sum of a sequence wherein every successive term contains a constant ratio to its preceding term. An Infinite geometric series has an infinite number of terms and can be represented as a, ar, ar2, ..., to ∞. Let us learn the infinite geometric series formula in the upcoming section.

What Is Infinite Geometric Series Formula?

The sum of the infinite geometric series formula of the infinite 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

Infinite Geometric Series Formula

The geometric series converges to a sum only if r < 1. If r> 1, the series does not converge and doesn't have a sum. For example 8, 12, 18, 27, .... is the given geometric series.

To find the sum : 8 + 12 + 18 + 27 + ..... , we find that a = 8 and r = 12/8 = 18/ 12 = 3/2

Here r > 1. Thus the sum does not converge and the series has no sum.

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

Example 1: Find the sum of the terms 1/9 + 1/27 + 1/81 + ... to ∞?

Solution: To find: Sum of the geometric series

Given:

a = 1/9, r = 1/3

Using the infinite geometric series formula,

Sn = a /(1-r)

Sn = (1/9)( 1 - 1/3)

Sn = 1/6

Answer: The sum of the given terms is 1/6

Example 2: Calculate the sum of series 1/5, 1/10, 1/20, .... if the series contains infinite terms.

Solution: To find: Sum of the geometric series

Given:

a = 1/5, r = 1/5

Using the infinite geometric series formula,

Sum = a /(1-r)

= (1/5)( 1 - 1/5)

= 1/4

Answer: The sum of the given terms is 1/4

Example 3: Find the sum of the geometric series 125, 25, 5, 1......∞

Solution: The series is,125+25+5+1+ ........ a =125, and = 25/125 = 1/5

Using the infinite geometric series formula,

Sum = a /(1-r)
= 125/(1-1/5)

= 125/(4/5)

= 625/4

Thus sum to infinity terms is

Answer: Thus sum to infinity in the given series 125+25+5+1+ ........ =& 625/4

The infinite geometric series formula is used to find the sum of all the terms in the geometric series without actually calculating them individually. The infinite geometric series formula is given as:

\(S_{n}=\dfrac{a}{1-r}\)

Where

• a is the first term
• r is the common ratio

A tangent of a circle in geometry is defined as a straight line that touches the circle at only one point.

Can the Sum of an Infinite Geometric Series be Negative?

The sum of an infinite series implies that the series is geometric and an infinite arithmetic series can never converge. So if the common ratio is positive there can be no negative sum.

What Is the Sum of Infinite GP?

The sum to infinite GP means, the sum of terms in an infinite GP. The infinite geometric series formula is S∞ = a/(1 – r), where a is the first term and r is the common ratio.

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