Find The Sum Of The Infinite Geometric Series Calculator

Calculate the Sum

First few terms and partial sums of the series

Term (n)	Term Value (a*r^(n-1))	Partial Sum (S_n)
Enter valid inputs to see the table.

What is the Sum of an Infinite Geometric Series?

The sum of an infinite geometric series is the value that the sum of the terms of a geometric series approaches as the number of terms goes to infinity. 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 an infinite geometric series with the first term 'a' and common ratio 'r', the sum exists (or converges) only if the absolute value of the common ratio is less than 1 (i.e., |r| < 1 or -1 < r < 1). If |r| ≥ 1, the series either diverges to infinity or oscillates, and it does not have a finite sum. Our Sum of an Infinite Geometric Series Calculator helps you find this sum when it exists.

This concept is useful in various fields like mathematics, physics, economics, and finance to model processes that involve repeated scaling or discounting over an infinite period.

Common misconceptions include assuming all infinite series have a sum, or that the sum is always very large. In fact, a convergent infinite geometric series has a finite sum, often surprisingly small.

Sum of an Infinite Geometric Series Formula and Mathematical Explanation

A geometric series is defined as a + ar + ar² + ar³ + … + ar^n-1 + …

The sum of the first n terms of a geometric series is given by:

S_n = a(1 – rⁿ) / (1 – r)

To find the sum of an infinite geometric series, we look at the limit of S_n as n approaches infinity (n → ∞).

If |r| < 1, then as n → ∞, rⁿ → 0. In this case, the formula for the sum of an infinite geometric series becomes:

S = lim_n→∞ S_n = lim_n→∞ [a(1 – rⁿ) / (1 – r)] = a(1 – 0) / (1 – r) = a / (1 – r)

So, the sum of an infinite geometric series with |r| < 1 is:

S = a / (1 – r)

If |r| ≥ 1, the term rⁿ does not approach 0 as n → ∞, and the series does not converge to a finite sum (it diverges).

Variables Table

Variable	Meaning	Unit	Typical Range
S	Sum of the infinite geometric series	Same as 'a'	Depends on 'a' and 'r'
a	First term of the series	Dimensionless or units	Any real number
r	Common ratio	Dimensionless	-1 < r < 1 (for convergence)
n	Term number (for partial sums)	Integer	1, 2, 3, …
Sn	Sum of the first n terms (Partial Sum)	Same as 'a'	Varies

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 + …

• First term (a) = 0.3
• Common ratio (r) = 0.03 / 0.3 = 0.1

Since |r| = |0.1| < 1, the series converges. Using the formula S = a / (1 - r):

S = 0.3 / (1 – 0.1) = 0.3 / 0.9 = 3/9 = 1/3

So, 0.333… is equal to 1/3. Our Sum of an Infinite Geometric Series Calculator would confirm this.

Example 2: Present Value of a Perpetuity

In finance, a perpetuity is a stream of equal payments that continues forever. If the payment is $100 per year and the discount rate is 5% (0.05), the present value of these future payments can be seen as an infinite geometric series where the first payment is received one year from now:

PV = 100/(1.05) + 100/(1.05)² + 100/(1.05)³ + …

• First term (a) = 100 / 1.05 ≈ 95.238
• Common ratio (r) = 1 / 1.05 ≈ 0.95238

Since |r| < 1, the sum is:

S = (100/1.05) / (1 – 1/1.05) = (100/1.05) / ((1.05-1)/1.05) = (100/1.05) / (0.05/1.05) = 100 / 0.05 = $2000

The present value of the perpetuity is $2000. This is a direct application of the geometric series sum formula.

How to Use This Sum of an Infinite Geometric Series Calculator

1. Enter the First Term (a): Input the initial value of your geometric series into the "First Term (a)" field.
2. Enter the Common Ratio (r): Input the common ratio into the "Common Ratio (r)" field. Remember, for the sum to be finite and calculable by the formula S = a/(1-r), the absolute value of 'r' must be less than 1 (-1 < r < 1). The calculator will check this.
3. View the Results: The calculator automatically updates and displays:
   • The Sum (S), if |r| < 1.
   • A message indicating whether the series converges or diverges.
   • The value of the denominator (1-r).
   • The formula used.
   • A table showing the first few terms and their partial sums.
   • A chart visualizing the partial sums approaching the total sum (if convergent).
4. Reset: Click the "Reset" button to clear the inputs and results and return to default values.
5. Copy Results: Click "Copy Results" to copy the main sum, intermediate values, and the convergence condition to your clipboard.

The Sum of an Infinite Geometric Series Calculator provides instant feedback, helping you understand how 'a' and 'r' influence the sum and convergence.

Key Factors That Affect the Sum of an Infinite Geometric Series

• First Term (a): The sum 'S' is directly proportional to 'a'. If you double 'a', the sum 'S' also doubles, provided 'r' remains the same and |r| < 1.
• Common Ratio (r): This is the most critical factor.
  • |r| < 1: The series converges, and a finite sum exists. The closer |r| is to 1, the larger the magnitude of the sum becomes (for a fixed 'a'). The closer |r| is to 0, the closer the sum is to 'a'.
  • |r| ≥ 1: The series diverges. If r ≥ 1 (and a ≠ 0), the sum goes to infinity (or negative infinity if a < 0 and r=1). If r ≤ -1, the terms oscillate with increasing or constant magnitude, and the sum does not approach a finite value.
• Sign of 'a' and 'r': The sign of 'a' determines the sign of the sum if 1-r is positive (i.e., r < 1). If r is negative, the terms of the series alternate in sign.
• Magnitude of 'r': Even within -1 < r < 1, the closer |r| is to 1, the more terms are needed for the partial sums to get very close to the final sum S.
• Convergence Condition: The absolute value of 'r' being less than 1 is the fundamental condition for the existence of a finite sum using the S = a/(1-r) formula.
• Nature of 'a' and 'r' (Real vs Complex): While this calculator deals with real numbers, 'a' and 'r' can also be complex numbers, leading to complex sums, provided |r| < 1.

Frequently Asked Questions (FAQ)

1. What happens if the common ratio |r| is equal to 1?

If r = 1 and a ≠ 0, the series is a + a + a + …, which diverges to ∞ (if a > 0) or -∞ (if a < 0). If r = -1 and a ≠ 0, the series is a - a + a - a + ..., which oscillates between a and 0, and does not converge. The formula S = a/(1-r) is undefined as the denominator becomes zero.

2. What happens if the common ratio |r| is greater than 1?

If |r| > 1, the terms ar^n-1 grow in magnitude as n increases, so the sum of the terms diverges to infinity or oscillates with increasing amplitude. No finite sum exists.

3. Can the first term 'a' be zero?

Yes. If a = 0, then all terms are zero, and the sum is 0, regardless of 'r'.

4. Can the common ratio 'r' be negative?

Yes. If -1 < r < 0, the series converges, and its terms alternate in sign. For example, if a=1 and r=-0.5, the series is 1 - 0.5 + 0.25 - 0.125 + ... and the sum is S = 1 / (1 - (-0.5)) = 1 / 1.5 = 2/3.

5. How is this calculator different from a finite geometric series calculator?

This Sum of an Infinite Geometric Series Calculator finds the sum of an infinite number of terms, which is only possible if |r|<1. A finite geometric series calculator sums a specific, finite number of terms (n), and works for any value of 'r' (as long as r≠1 for the formula Sn=a(1-r^n)/(1-r)).

6. What are some real-world applications of the sum of an infinite geometric series?

Applications include calculating the present value of perpetuities in finance, modeling the total distance a bouncing ball travels, understanding repeating decimals, and in some physics problems involving waves or oscillations.

7. How accurate is the sum calculated?

The formula S = a / (1 – r) gives the exact sum if |r| < 1. The calculator performs standard floating-point arithmetic, so the precision is very high.

8. Does the chart show the actual infinite sum?

The chart shows the partial sums for the first few terms, illustrating how they approach the total sum 'S' calculated by the formula. It visually represents the convergence for a limited number of terms.

Related Tools and Internal Resources

• Finite Geometric Series Calculator: Calculate the sum of the first 'n' terms of a geometric series.
• Arithmetic Series Calculator: Find the sum of an arithmetic series.
• Sequence Calculator: Analyze various types of sequences.
• Limit Calculator: Evaluate limits of functions, relevant to infinite series convergence.
• Derivative Calculator: Find derivatives of functions.
• Integral Calculator: Calculate definite and indefinite integrals.