Find Infinite Sum Calculator

Calculate the Infinite Sum

First 10 terms and partial sums of the series.

Term Number (n)	Term Value (a*r^(n-1))	Partial Sum (Sn)

Understanding the Infinite Geometric Series Sum Calculator

What is an Infinite Geometric Series Sum Calculator?

An infinite geometric series sum calculator is a tool used to determine the sum of an infinite number of terms that follow a geometric progression, provided the series converges. A geometric series is a sequence of numbers 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 calculator determines if the sum approaches a finite value (converges) or grows indefinitely (diverges).

This calculator is particularly useful for students of mathematics (algebra, pre-calculus, calculus), engineers, physicists, and economists who deal with processes or models that can be represented by infinite geometric series, such as compound interest over infinite periods (conceptually), or the total distance traveled by a bouncing ball.

A common misconception is that all infinite series have an infinite sum. However, if the absolute value of the common ratio |r| is less than 1, the terms get progressively smaller, and the sum approaches a finite limit. Our infinite geometric series sum calculator helps identify this limit.

Infinite Geometric Series Sum Formula and Mathematical Explanation

A geometric series is defined by its first term, 'a', and its common ratio, 'r'. The terms are a, ar, ar², ar³, …

The sum of the first 'n' terms of a geometric series (a finite sum, S_n) is given by:

S_n = a(1 – rⁿ) / (1 – r)

For an infinite geometric series, we consider what happens as n approaches infinity (n → ∞). If the absolute value of the common ratio |r| < 1, then rⁿ approaches 0 as n → ∞. In this case, the series converges, and the infinite sum (S_∞) is:

S_∞ = a / (1 – r)

If |r| ≥ 1, the terms either do not decrease to zero or their absolute values increase, and the series diverges, meaning the sum does not approach a finite value (it goes to infinity or oscillates without limit). The infinite geometric series sum calculator checks this condition.

Variables Table:

Variable	Meaning	Unit	Typical Range for Convergence
a	The first term of the series	Unitless or units of the term	Any real number
r	The common ratio	Unitless	-1 < r < 1
S∞	The sum of the infinite geometric series	Same as 'a'	Finite value if |r| < 1
n	Term number (for partial sums)	Integer	1, 2, 3, …

Practical Examples (Real-World Use Cases)

Example 1: The Bouncing Ball

A ball is dropped from a height of 10 meters. Each time it hits the ground, it bounces back to 60% (0.6) of its previous height. What is the total vertical distance traveled by the ball before it comes to rest?

The initial downward distance is 10m. Then it bounces up 10*0.6 = 6m and down 6m, then up 6*0.6 = 3.6m and down 3.6m, and so on.

Total distance = 10 (initial) + [2 * (10*0.6) + 2 * (10*0.6²) + 2 * (10*0.6³) + …]

The part in the brackets is 2 times an infinite geometric series with a = 10*0.6 = 6 and r = 0.6. Sum of bounces = 2 * [6 / (1 – 0.6)] = 2 * [6 / 0.4] = 2 * 15 = 30 meters.

Total distance = 10 + 30 = 40 meters. Using the infinite geometric series sum calculator with a=6 and r=0.6 gives 15, multiplied by 2 and adding initial 10 gives 40.

Example 2: Repeating Decimals

Express the repeating decimal 0.777… as a fraction.

0.777… = 0.7 + 0.07 + 0.007 + … This is an infinite geometric series with a = 0.7 and r = 0.1.

Using the formula S_∞ = a / (1 – r) = 0.7 / (1 – 0.1) = 0.7 / 0.9 = 7/9. You can verify this using the infinite geometric series sum calculator with a=0.7 and r=0.1.

How to Use This Infinite Geometric Series Sum Calculator

1. Enter the First Term (a): Input the very first number in your geometric series into the "First Term (a)" field.
2. Enter the Common Ratio (r): Input the ratio between any term and its preceding term into the "Common Ratio (r)" field. For the sum to be finite (converge), r must be between -1 and 1 (exclusive).
3. Calculate: Click the "Calculate Sum" button or simply change the input values. The calculator will automatically update.
4. Review Results: The calculator will display:
   • The Infinite Sum (S_∞) if the series converges (|r| < 1).
   • A message indicating "Diverges" if the series does not converge (|r| ≥ 1).
   • The values of 'a' and 'r' you entered.
   • The convergence status.
5. Examine Table and Chart: If the series converges, a table showing the first few terms and their partial sums, and a chart visualizing the partial sums approaching the infinite sum, will be displayed.
6. Reset: Click "Reset" to clear the inputs and results and return to default values.
7. Copy: Click "Copy Results" to copy the main result and inputs to your clipboard.

The infinite geometric series sum calculator provides immediate feedback on whether your series has a finite sum and what that sum is.

Key Factors That Affect Infinite Geometric Series Sum Results

1. First Term (a): The starting value of the series. The sum is directly proportional to 'a'. If you double 'a', you double the sum (assuming 'r' remains the same and |r| < 1).
2. Common Ratio (r): This is the most critical factor.
   • If |r| < 1, the series converges, and a finite sum exists. The closer |r| is to 0, the faster the series converges, and the smaller the sum relative to 'a'/(1-|r|).
   • If |r| ≥ 1, the series diverges, and the sum is infinite or undefined as a single finite number. The infinite geometric series sum calculator will indicate this.
3. Sign of 'r': If 'r' is positive, all terms have the same sign as 'a'. If 'r' is negative, the terms alternate in sign, but the sum still converges if |r| < 1.
4. Magnitude of 'r' close to 1: When |r| is very close to 1 (but less than 1), the denominator (1-r) is very small, leading to a large sum (S_∞). The convergence is also slower.
5. Number of Terms (in concept): Although we are calculating the sum to infinity, understanding how quickly the partial sums approach the infinite sum is related to |r|. Smaller |r| means faster convergence.
6. Context of the Problem: Whether 'a' and 'r' represent distances, money, probabilities, or other quantities will determine the units and interpretation of the sum.

Understanding these factors is crucial for interpreting the results of the infinite geometric series sum calculator correctly.

Frequently Asked Questions (FAQ)

What is a geometric series?

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

When does an infinite geometric series have a finite sum?

An infinite geometric series has a finite sum (converges) only when the absolute value of the common ratio 'r' is less than 1 (i.e., -1 < r < 1).

What if the common ratio |r| is 1 or greater?

If |r| ≥ 1, the series diverges. This means the sum of the terms does not approach a single finite value; it either goes to positive or negative infinity or oscillates without settling down.

How do I find the common ratio 'r'?

Divide any term by its preceding term. For example, if the series is 2, 4, 8, …, r = 4/2 = 2. If it's 9, 3, 1, …, r = 3/9 = 1/3.

Can the first term 'a' be zero?

Yes, if 'a' is zero, all terms are zero, and the sum is zero, regardless of 'r'. Our infinite geometric series sum calculator handles this.

Can 'r' be negative?

Yes, if 'r' is negative and |r| < 1, the series converges, and the terms alternate in sign (e.g., 8, -4, 2, -1,...).

What does it mean for a series to converge?

It means that as you add more and more terms, the sum of those terms gets closer and closer to a specific finite number, which is the infinite sum S_∞.

Is the sum always accurate?

The formula S_∞ = a / (1 – r) gives the exact sum if |r| < 1. The infinite geometric series sum calculator uses this exact formula.