Geometric Series Calculator: How to Find the Sum to Infinity and Partial Sums
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
A geometric series calculator can tell you something that sounds impossible: a ball dropped from 8 feet, rebounding to three-quarters of its height on every bounce, travels exactly 56 feet before it stops. Not “about 56” — exactly 56, even though it bounces infinitely many times. That’s the strange gift of geometric series: infinitely many additions, one finite answer, and a formula short enough to fit on a fingernail. This article covers the two formulas that answer, where a/(1 − r) actually comes from, and the real-world places — repeating decimals, drug dosing, perpetuities — where the sum to infinity quietly runs the numbers.
The Two Formulas Every Geometric Series Calculator Runs
A geometric series adds terms that each equal the previous term times a fixed number: a + ar + ar² + ar³ + … The first term is a, and r is the common ratio — find it by dividing any term by the one before it. Everything the calculator above reports comes from two closed forms:
Partial sum: S(N) = a(1 − rᴺ)/(1 − r) (any r ≠ 1)
Sum to infinity: S = a/(1 − r) (only when |r| < 1)
The condition |r| < 1 is the whole game. When the ratio sits strictly between −1 and 1, each term is a fixed fraction of the last, the terms die off geometrically, and the sum settles. At r = 0.75 the series 8 + 6 + 4.5 + … lands on 8/(1 − 0.75) = 32. Nudge the ratio to 1.05 and the same formula is meaningless — the terms grow 5% each step and the total runs away. There’s no partial credit at the boundary either: r = 1 and r = −1 both diverge, just in different styles.
Where a/(1 − r) Comes From: Multiply, Shift, Subtract
The infinite-sum formula isn’t handed down from anywhere mysterious — it falls out of one algebraic move. Call the sum S and multiply the whole thing by r:
S = a + ar + ar² + ar³ + …
rS = ar + ar² + ar³ + …
S − rS = a (every other term cancels in pairs)
S(1 − r) = a → S = a/(1 − r)
Multiplying by r shifts the series one slot to the right, so subtraction wipes out everything except the first term. The same trick on the first N terms leaves a − arN instead of just a, which is where S(N) = a(1 − rᴺ)/(1 − r) comes from. And it exposes exactly why |r| ≥ 1 breaks things: the argument silently assumes the leftover rᴺ term vanishes as N grows. For |r| < 1 it does. For r = 2 it explodes, and for r = −1 it flips between +1 and −1 forever — which is why Grandi’s series 1 − 1 + 1 − 1 + … has no sum, no matter how tempting the answer “½” looks.
Bouncing Balls, Drug Doses, and Why 0.999… = 1
Start with the ball from the opening. It falls 8 feet, then travels each bounce height twice — up and back down. The bounce heights are 6, 4.5, 3.375, … — a geometric series with a = 6 and r = 3/4, which sums to 6/(1 − 3/4) = 24 feet. Total distance: 8 + 2 × 24 = 56 feet. Zeno’s dichotomy paradox dissolves the same way: crossing half a room, then half the remainder, and so on is just ½ + ¼ + ⅛ + … = 1. Infinitely many steps, finite distance.
Repeating decimals are geometric series in disguise. The decimal 0.999… means 0.9 + 0.09 + 0.009 + …, which is a = 0.9, r = 0.1, so S = 0.9/0.9 = 1. Not approximately 1 — equal to 1. The same expansion turns 0.727272… into 0.72/(1 − 0.01) = 72/99 = 8/11, which is exactly what our repeating decimal to fraction calculator does under the hood.
Two more places the formula earns money. Pharmacology: take 250 mg of a drug every 24 hours while 40% of the previous amount is still in your system at each dose. The level right after dose N is 250(1 + 0.4 + 0.4² + …), which climbs toward a ceiling of 250/0.6 ≈ 416.7 mg — the steady state doctors design dosing schedules around. Finance: a perpetuity paying $1,000 a year forever, discounted at 5%, is worth (1000/1.05)/(1 − 1/1.05) = $20,000 today. An infinite stream of payments, priced with one division.
Reading a and r out of Sigma Notation
Homework rarely says “a = 6/5, r = 2/5.” It says Σ 3(2/5)ⁿ from n = 1 to ∞, and the most common wrong answer comes from misreading the first term. The first term is whatever the expression equals at the starting index — here n = 1 gives a = 3 × (2/5) = 6/5, not 3. The sum is (6/5)/(1 − 2/5) = (6/5)/(3/5) = 2. Had the index started at n = 0, the answer would be 3/(3/5) = 5 instead. Same expression, different sum — the lower bound of the sigma matters as much as the formula. If sigma notation itself is the sticking point, the sigma notation calculator expands any Σ expression term by term.
Messier-looking sums usually just need one rewrite. Take Σ 4ⁿ⁺¹/7ⁿ from n = 0: split the numerator as 4 · 4ⁿ to get 4(4/7)ⁿ, so a = 4, r = 4/7, and S = 4/(3/7) = 28/3 ≈ 9.33. The test for “is this geometric at all” is always the same — divide consecutive terms and check the answer is constant. If the ratio depends on n, you’re holding a different series, and a general summation calculator is the better tool.
Five Series Worth Recognizing on Sight
| Series | a | r | Sum |
|---|---|---|---|
| ½ + ¼ + ⅛ + … (Zeno) | 1/2 | 1/2 | 1 |
| 0.9 + 0.09 + 0.009 + … (0.999…) | 0.9 | 0.1 | 1 |
| 1 − ½ + ¼ − ⅛ + … | 1 | −1/2 | 2/3 |
| 1 − 1 + 1 − 1 + … (Grandi) | 1 | −1 | diverges |
| 1 + 1.05 + 1.05² + … | 1 | 1.05 | diverges |
The third row deserves a second look. A negative ratio makes the partial sums leapfrog the limit — 1, then 0.5, then 0.75, then 0.625 — landing alternately above and below 2/3. Convergence doesn’t care: |−½| = ½ < 1 is all that’s checked. Punch a = 1, r = −1/2 into the calculator and watch the “distance from the limit” column halve with every term while the sums zigzag.
Series or Sequence? Base or Exponent? The Two Mix-Ups That Cost Points
First mix-up: a geometric sequence is the list 8, 6, 4.5, 3.375, …; a geometric series is what you get when you add that list. The distinction changes the question entirely — the sequence above converges to 0, while its series converges to 32. If you need the 15th term or the ratio between given terms rather than a sum, the geometric sequence calculator handles the list side of the problem.
Second mix-up: geometric series versus p-series. In Σ (1/2)ⁿ the variable lives in the exponent; in Σ 1/n² it lives in the base. They look like cousins and obey completely different rules — geometric series converge when |r| < 1, p-series when p > 1, and mixing up the two conditions is a classic exam giveaway. A quick self-check: apply the ratio test to a geometric series and the limit comes out to exactly |r|, which is why the geometric case is the one series family where convergence is instant to decide. For anything that isn’t geometric — ratios that drift with n, factorials, powers of n — run it through the series convergence calculator and let it pick the right test.
