Find The Sum Of The Infinite Series Calculator

First few terms and partial sums of the series.

n (Term #)	Term Value (ar^n-1)	Partial Sum (Sn)

What is the Sum of an Infinite Geometric Series Calculator?

A Sum of Infinite Geometric Series Calculator is a tool used to find the sum of a geometric series that goes on forever (has an infinite number of terms). 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). For the sum to be finite and calculable, the series must be convergent, which means the absolute value of the common ratio |r| must be less than 1.

This calculator is useful for students studying mathematics, particularly calculus and series, as well as engineers, physicists, and economists who encounter such series in their work. It helps quickly determine the sum if the series converges or indicate if it diverges.

A common misconception is that all infinite series have an infinite sum. However, if the terms decrease rapidly enough (as in a convergent geometric series), the sum can be a finite value. The Sum of Infinite Geometric Series Calculator helps distinguish between these cases based on the common ratio.

Sum of Infinite Geometric Series 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, and so on.

The sum of the first 'n' terms of a geometric series (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. If the absolute value of the common ratio |r| is less than 1 (-1 < r < 1), then as n becomes very large, rⁿ approaches 0. In this case, the series converges, and the sum to infinity (S) is:

S = a / (1 – r)

If |r| ≥ 1, the terms either do not decrease to zero or their magnitudes increase, and the series diverges, meaning the sum is not a finite number (or it oscillates without approaching a limit).

Variables Table

Variable	Meaning	Unit	Typical Range
a	First term of the series	Unitless (or same as terms)	Any real number
r	Common ratio	Unitless	Any real number (but sum converges only for -1 < r < 1)
S	Sum of the infinite series	Unitless (or same as terms)	Finite if |r| < 1, otherwise undefined/infinite
Sn	Sum of the first n terms	Unitless (or same as terms)	Varies

Our Sum of Infinite Geometric Series Calculator uses the formula S = a / (1 – r) when |r| < 1.

Practical Examples (Real-World Use Cases)

Example 1: Repeating Decimals

Consider the repeating decimal 0.777… We can express this as an infinite geometric series:

0.777… = 0.7 + 0.07 + 0.007 + … = 7/10 + 7/100 + 7/1000 + …

Here, the first term a = 7/10, and the common ratio r = (7/100) / (7/10) = 1/10 = 0.1.

Since |r| = 0.1 < 1, the series converges. Using the Sum of Infinite Geometric Series Calculator or the formula:

S = a / (1 – r) = (7/10) / (1 – 1/10) = (7/10) / (9/10) = 7/9.

So, 0.777… = 7/9.

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?

Initial drop: 10 m

First bounce up: 10 * 0.6 = 6 m, then down 6 m (total 12 m)

Second bounce up: 6 * 0.6 = 3.6 m, then down 3.6 m (total 7.2 m)

And so on. The distances traveled up and down after the initial drop form two identical geometric series: 6 + 3.6 + 2.16 + …

For this series, a = 6 and r = 0.6. The sum is S = 6 / (1 – 0.6) = 6 / 0.4 = 15 m.

So, the ball travels 15 m up and 15 m down after the first drop, plus the initial 10 m drop.

Total distance = 10 + 15 + 15 = 40 meters. Our Sum of Infinite Geometric Series Calculator can find the sum of the upward or downward series (15m).

How to Use This Sum of Infinite Geometric Series Calculator

1. Enter the First Term (a): Input the very first number in your 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. Remember, for a finite sum, the absolute value of 'r' must be less than 1.
3. Calculate: The calculator automatically updates the results as you type, or you can click "Calculate Sum".
4. Read the Results:
   • Primary Result: Shows the sum of the infinite series if it converges. If |r| ≥ 1, it will indicate divergence.
   • Intermediate Values: Shows the convergence condition (|r| < 1 or |r| ≥ 1) and the value of (1-r).
   • Formula Explanation: Reminds you of the formula used (S = a / (1-r)).
5. Analyze Chart and Table: The chart visually represents the partial sums approaching the total sum (if convergent), and the table shows the values of the first few terms and their cumulative sums.
6. Reset: Click "Reset" to clear the fields and start over with default values.

Using the Sum of Infinite Geometric Series Calculator helps you quickly determine if a series converges and, if so, its sum, without manual calculation.

Key Factors That Affect Sum of Infinite Geometric Series Results

• First Term (a): The sum is directly proportional to 'a'. If 'a' doubles, the sum doubles, provided 'r' remains the same and the series converges.
• Common Ratio (r): This is the most critical factor.
  • If |r| < 1 (-1 < r < 1), the series converges to a finite sum. The closer |r| is to 0, the faster the convergence and the smaller the sum relative to |a/(1-r)|.
  • If |r| ≥ 1 (r ≥ 1 or r ≤ -1), the series diverges, and there is no finite sum. The Sum of Infinite Geometric Series Calculator will indicate this.
• Sign of 'a' and 'r': The signs of 'a' and 'r' determine the sign of the terms and the sum. If 'r' is negative, the terms alternate in sign.
• Magnitude of 'r' close to 1: As |r| gets very close to 1 (but still less than 1), the denominator (1-r) or (1+r if r is near -1) gets very small, leading to a large sum magnitude.
• Value of 'a' being zero: If the first term 'a' is 0, then all terms are 0, and the sum is 0, regardless of 'r'.
• Practical Precision: In real-world applications using the Sum of Infinite Geometric Series Calculator, the number of terms considered practically might be finite, even if the theoretical model is infinite.

Frequently Asked Questions (FAQ)

What happens if the common ratio |r| is greater than or equal to 1?

If |r| ≥ 1, the infinite geometric series diverges. This means the sum of its terms does not approach a finite value. If r=1 (and a≠0), the terms are constant, and the sum goes to infinity. If r > 1, the terms grow, and the sum goes to infinity. If r ≤ -1, the terms oscillate with increasing or constant magnitude, and the sum does not converge. Our Sum of Infinite Geometric Series Calculator will indicate divergence.

Can the common ratio 'r' be negative?

Yes, 'r' can be negative. If -1 < r < 0, the series converges, and its terms alternate in sign (e.g., a, -ar, ar², -ar³, …). The sum S = a / (1 – r) will still be finite.

What if the first term 'a' is zero?

If a = 0, every term in the series is zero (0, 0, 0, …), and the sum is 0, regardless of the value of 'r'.

How does the Sum of Infinite Geometric Series Calculator handle |r| = 1?

If you enter r=1 or r=-1, the calculator will indicate that the series diverges because the condition |r| < 1 is not met.

Is the sum always positive?

No. The sign of the sum S = a / (1 – r) depends on the signs of 'a' and (1 – r). If 'a' is positive and 1-r is positive (i.e., r < 1), the sum is positive. If 'a' is positive and 1-r is negative (not possible for convergence), or if 'a' is negative and 1-r is positive, the sum is negative.

Where is the formula S = a / (1 – r) derived from?

It comes from the formula for the sum of the first n terms, S_n = a(1 – rⁿ) / (1 – r). As n approaches infinity, if |r| < 1, then rⁿ approaches 0, leaving S = a(1 – 0) / (1 – r) = a / (1 – r).

Can I use this Sum of Infinite Geometric Series Calculator for finite series?

No, this calculator is specifically for infinite series where |r| < 1. For a finite number of terms, you would use the formula S_n = a(1 – rⁿ) / (1 – r).

What are some applications of infinite geometric series?

They appear in calculating the present value of perpetual annuities, fractal geometry (like the Koch snowflake), modeling repeating processes, and representing repeating decimals as fractions, as shown in our examples using the Sum of Infinite Geometric Series Calculator.

Related Tools and Internal Resources

• Arithmetic Sequence Calculator: Find terms in an arithmetic sequence.
• Geometric Sequence Calculator: Calculate terms of a geometric sequence.
• Series Convergence Test Calculators: Explore various tests for series convergence.
• Finite Geometric Series Sum Calculator: Calculate the sum of the first n terms.
• Repeating Decimal to Fraction Converter: Convert repeating decimals using geometric series concepts.
• Present Value of Perpetuity Calculator: An application of infinite geometric series in finance.