Sum of Infinite Geometric Series Calculator | Calculate S = a / (1-r)

Sum of Infinite Geometric Series Calculator

Calculate the Sum

Enter the first term (a) and the common ratio (r) to find the sum of an infinite geometric series, provided |r| < 1.

First Term (a):

The initial value of the series. Can be any real number.

Common Ratio (r):

The constant factor between successive terms. Must be between -1 and 1 (exclusive) for the sum to converge.

Results:

Enter values and calculate

Condition for Convergence (|r| < 1): –

Absolute Value of r (|r|): –

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Chart showing the first few terms (blue) and partial sums (green) approaching the infinite sum (red line, if convergent).

What is the Sum of an Infinite Geometric Series?

An infinite geometric series is a series (sum of terms of a sequence) where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). The sum of an infinite geometric series exists (i.e., it converges to a finite value) only if the absolute value of the common ratio is less than 1 ( |r| < 1 ). If |r| ≥ 1, the series either diverges to infinity or oscillates, and it does not have a finite sum.

This sum of infinite geometric series calculator helps you find this sum when the series converges.

It’s used in various fields like mathematics, physics, economics, and engineering to model phenomena that decrease or grow by a constant factor over time or steps, but only when the factor leads to convergence.

A common misconception is that any infinite series with decreasing terms will have a finite sum. This is not true; for a geometric series, the common ratio’s magnitude must be strictly less than 1.

Sum of Infinite Geometric Series Formula and Mathematical Explanation

The formula to find the sum (S) of an infinite geometric series is:

S = a / (1 – r)

Where:

S is the sum of the infinite series.
a is the first term of the series.
r is the common ratio.

This formula is derived from the formula for the sum of the first n terms of a geometric series, S_n = a(1 – rⁿ) / (1 – r). As n approaches infinity, if |r| < 1, then rⁿ approaches 0. Thus, S_n approaches a(1 – 0) / (1 – r) = a / (1 – r).

The condition |r| < 1 is crucial. If |r| ≥ 1, the rⁿ term does not approach 0, and the sum does not converge to a finite value.

Variables Table:

Variable	Meaning	Unit	Typical Range
S	Sum of the infinite geometric series	(Same as ‘a’)	Real number or undefined if \|r\| ≥ 1
a	First term	Varies	Any real number
r	Common ratio	Dimensionless	-1 < r < 1 (for convergence)

Practical Examples (Real-World Use Cases)

Example 1: Repeating Decimals

Consider the repeating decimal 0.333… This can be written as an infinite geometric series: 0.3 + 0.03 + 0.003 + …

Here, the first term (a) = 0.3, and the common ratio (r) = 0.03 / 0.3 = 0.1.

Since |r| = |0.1| < 1, the sum converges.

Using the formula S = a / (1 – r) = 0.3 / (1 – 0.1) = 0.3 / 0.9 = 3/9 = 1/3.

Our sum of infinite geometric series calculator would confirm this.

Example 2: Bouncing Ball

A ball is dropped from a height of 10 meters. Each time it bounces, it reaches 60% of its previous height. What is the total vertical distance traveled by the ball before it comes to rest?

The distances it falls are: 10, 10*(0.6), 10*(0.6)², …

The distances it rises are: 10*(0.6), 10*(0.6)², …

Total distance = 10 (initial drop) + 2 * [10*(0.6) + 10*(0.6)² + …]

The series in the brackets is geometric with a = 10*(0.6) = 6 and r = 0.6. |0.6| < 1.

Sum of the rise/fall series = 6 / (1 – 0.6) = 6 / 0.4 = 15 meters.

Total distance = 10 + 2 * 15 = 10 + 30 = 40 meters. You could use the sum of infinite geometric series calculator for the part inside the bracket.

How to Use This Sum of Infinite Geometric Series Calculator

Enter the First Term (a): Input the initial value of your series into the “First Term (a)” field.
Enter the Common Ratio (r): Input the common ratio into the “Common Ratio (r)” field. Remember, for the sum to be finite, |r| must be less than 1.
Calculate: The calculator automatically updates as you type, or you can click “Calculate”.
Read the Results:
- Primary Result: Shows the sum ‘S’ if |r| < 1, or indicates divergence if |r| ≥ 1.
- Intermediate Results: Shows if the convergence condition is met and the absolute value of r.
- Formula Explanation: Reminds you of the formula used.
Analyze the Chart: The chart visualizes the terms and partial sums, helping you see how the series converges (or diverges).
Reset: Click “Reset” to clear the fields to default values.
Copy Results: Click “Copy Results” to copy the main sum and conditions to your clipboard.

Use the sum of infinite geometric series calculator to quickly verify sums or explore the behavior of different series.

Key Factors That Affect Sum of Infinite Geometric Series Results

Value of the First Term (a): The sum S is directly proportional to ‘a’. If ‘a’ doubles, S doubles, provided ‘r’ remains constant and |r| < 1.
Value of the Common Ratio (r): This is the most critical factor. The sum only exists if -1 < r < 1. The closer |r| is to 1 (but still less than 1), the larger the magnitude of the sum (for a given 'a'). The closer r is to 0, the faster the terms decrease, and the sum is closer to 'a'.
Condition |r| < 1: If |r| is greater than or equal to 1, the series does not have a finite sum. It diverges. The calculator will indicate this.
Sign of ‘a’ and ‘r’: The sign of ‘a’ determines the overall sign of the sum if 1-r is positive. If r is negative, the terms alternate in sign.
Magnitude of |r| relative to 1: The closer |r| gets to 1, the slower the convergence, and the more terms are needed to get close to the infinite sum.
Zero values: If a=0, the sum is 0 (trivial case). If r=0, the sum is just ‘a’.

Understanding these factors is key to interpreting the output of the sum of infinite geometric series calculator.

Frequently Asked Questions (FAQ)

What happens if the common ratio (r) is 1 or -1?: If r = 1 (and a ≠ 0), the series is a + a + a + …, which diverges to infinity (or -infinity if a < 0). If r = -1, the series is a - a + a - a + ..., which oscillates between a and 0 and does not converge to a single sum. The sum of infinite geometric series calculator will show it diverges.
What if the absolute value of r is greater than 1?: If |r| > 1, the terms of the series grow in magnitude, and the sum diverges to infinity (or -infinity). It does not have a finite sum.
Can the first term ‘a’ be zero?: Yes. If ‘a’ is 0, every term in the series is 0, and the sum is 0, regardless of ‘r’.
Can ‘a’ or ‘r’ be negative?: Yes, both ‘a’ and ‘r’ can be negative. If ‘r’ is negative (and |r| < 1), the terms will alternate in sign, but the series will still converge.
How many terms are needed for the partial sum to be close to the infinite sum?: This depends on how close |r| is to 1. The closer |r| is to 1, the more terms you need. The closer |r| is to 0, the fewer terms you need.
Is this calculator the same as a geometric sequence calculator?: No, a geometric sequence calculator deals with the terms of the sequence (a, ar, ar², …), while this sum of infinite geometric series calculator finds the sum of all these terms when the series converges.
Where is the formula S = a / (1 – r) derived from?: It comes from the limit of the sum of the first n terms, S_n = a(1 – rⁿ) / (1 – r), as n approaches infinity, given |r| < 1, where rⁿ approaches 0.
Can I use this calculator for financial calculations like annuities?: While some financial concepts involve geometric series (like present value of an infinite stream of payments with growth), they often involve more complex formulas. This calculator is for the pure mathematical concept, but you can see the connection if the growth rate is your ‘r’. For specific financial tools, see our calculus calculators or general math section.

Find The Sum Of An Infinite Geometric Series Calculator