Sum of Geometric Series Calculator

Calculate the sum of a geometric series with our free online calculator. Supports both finite series and infinite convergent series with step-by-step solutions.

Infinite Series (|r| < 1)

First Term (a₁):

Common Ratio (r):

Number of Terms (n):

Examples:

Understanding Geometric Series

A geometric series is the sum of the terms of a geometric sequence. While a sequence lists the numbers (2, 6, 18, 54...), a series adds them together (2 + 6 + 18 + 54 + ...). Geometric series appear in finance (present value of annuities), physics (total distance of bouncing ball), and pure mathematics.

Infinite geometric series are fascinating – they can sum to a finite value even though they have infinitely many terms. This happens when the common ratio is between -1 and 1, causing terms to shrink toward zero fast enough that the sum converges.

Geometric Series Formulas

Finite Series Sum

Sₙ = a₁(1 - r^n) / (1 - r), for r ≠ 1

Sum of the first n terms. When r = 1, all terms equal a₁, so Sₙ = n × a₁.

Infinite Series Sum

S∞ = a₁ / (1 - r), for |r| < 1

Only converges when the absolute value of r is less than 1. Otherwise, the sum diverges.

Worked Examples

Example 1: Finite Series (r = 0.5)

Find: 1 + 1/2 + 1/4 + 1/8 + ... (10 terms)

a₁ = 1, r = 0.5, n = 10

S₁₀ = 1 × (1 - 0.5^10) / (1 - 0.5)

S₁₀ = (1 - 0.0009766) / 0.5

S₁₀ = 0.9990234 / 0.5 = 1.998

Approaches 2 as n increases

Example 2: Infinite Series

Find: 1 + 1/2 + 1/4 + 1/8 + ... (forever)

a₁ = 1, r = 0.5

|r| = 0.5 < 1 ✓ (converges)

S∞ = 1 / (1 - 0.5) = 1 / 0.5 = 2

The infinite sum equals exactly 2!

Example 3: Growing Series (r = 3)

Find: 2 + 6 + 18 + 54 + 162 + 486 (6 terms)

a₁ = 2, r = 3, n = 6

S₆ = 2 × (1 - 3^6) / (1 - 3)

S₆ = 2 × (1 - 729) / (-2)

S₆ = 2 × (-728) / (-2) = 728

Verify: 2+6+18+54+162+486 = 728 ✓

Example 4: Zeno's Paradox

To walk across a room, you must first go halfway, then half of remaining, etc.

1/2 + 1/4 + 1/8 + 1/16 + ...

a₁ = 0.5, r = 0.5

S∞ = 0.5 / (1 - 0.5) = 0.5 / 0.5 = 1

You DO reach the other side! The infinite sum equals 1 (the whole distance).

Quick Fact

Archimedes used geometric series around 250 BCE to calculate the area of a parabola. He showed that the area is 4/3 times the area of a certain triangle – essentially summing an infinite geometric series centuries before the formal concept existed.

Frequently Asked Questions

Why does the infinite series only work for |r| < 1?

When |r| ≥ 1, terms don't shrink – they stay the same size or grow. Adding infinitely many non-shrinking terms gives infinity. When |r| < 1, terms approach zero fast enough that the sum converges to a finite value.

What if r is negative?

For infinite series, we need |r| < 1, so -1 < r < 1. With negative r, terms alternate signs but still converge. For example, 1 - 1/2 + 1/4 - 1/8 + ... = 1/(1-(-0.5)) = 2/3.

How is this used in finance?

Present value calculations use geometric series. If you receive $100 yearly forever (a perpetuity) and discount at 5%, the present value is $100/0.05 = $2000. This is an infinite geometric series.

What's the connection to repeating decimals?

Repeating decimals are geometric series! 0.333... = 3/10 + 3/100 + 3/1000 + ... = (3/10)/(1-1/10) = 3/9 = 1/3. Every repeating decimal equals a fraction.

Can I find the sum starting from a term other than the first?

Yes. Find which term you're starting from, treat it as your new a₁, and adjust n accordingly. Or calculate the full sum and subtract the terms you don't want.

What happens when r = 1?

Every term equals a₁. The finite sum is n × a₁. The infinite series diverges (goes to infinity) unless a₁ = 0. The standard formula doesn't work because it divides by (1-r) = 0.

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