Infinite Geometric Series Calculator
This infinite geometric series calculator finds the sum to infinity of a series that never ends — when there is one. Give the first term and the common ratio, and it tests the ratio first, then works the total out and shows the running sum closing in on it.
The famous example: 1 + ½ + ¼ + ⅛ + … Add forever and you get exactly 2. Not nearly 2 — 2. The running totals are 1, 1½, 1¾, 1⅞, and the gap left to close halves every single step: 1, ½, ¼, ⅛. It never becomes nothing, and it never stops shrinking, and that is what the total means.
When Adding Forever Gives an Answer
This is the whole question, and it comes before any arithmetic: an infinite geometric series has a total exactly when
|r| < 1 — the ratio lies strictly between −1 and 1.
When that holds, the series is said to converge: each term is a fixed fraction of the one before, so the terms fall towards nothing quickly enough that the running total closes in on a value rather than running away. Then, and only then:
S = a₁ ÷ (1 − r)
For 1 + ½ + ¼ + … that is 1 ÷ (1 − ½) = 1 ÷ ½ = 2. For 2 + ⅔ + ⅙ + … it is 2 ÷ ⅔ = 3.
The trap is that the formula happily produces a number outside its own condition. Put r = 2 into it and you get a₁ ÷ (−1), a perfectly ordinary-looking answer for a series whose terms are doubling. It is meaningless. Check the ratio first — this calculator does, and it prints that check as its first line rather than burying it.
The Three Ways It Fails
When the ratio is not inside the window, most sources say the series diverges and stop. That single word covers three genuinely different things, and knowing which one you are looking at is worth more than the label.
|r| > 1 — it runs away. Each term is bigger than the last, so the running total passes any number you care to name. 1 + 2 + 4 + 8 + … has no total, and no number is even close.
r = 1 — it also runs away, but for a different reason. Here every term is the same: 5 + 5 + 5 + 5 + … The terms are not growing at all, yet the total still climbs by 5 forever. The formula cannot help either, because 1 − r is zero and it would be dividing by nothing.
r = −1 — it does not run away at all. This one is different in kind. The series 1 − 1 + 1 − 1 + … has running totals of 1, 0, 1, 0, 1, 0 — forever. It never grows, and it never settles on anything. Calling that "diverging" is technically right and practically misleading: nothing is running off anywhere. It simply refuses to land.
This page names which of the three you have hit, in its own words, rather than handing you one verdict for all of them.
Watching the Total Close In
The idea that you can add endlessly many numbers and land on a finite one is the part people do not believe, and no amount of formula will fix that. Seeing it will.
For 1 + ½ + ¼ + … the calculator lists the running total after each term, and beside each one the gap still left to the total:
1 — gap 1
1½ — gap ½
1¾ — gap ¼
1⅞ — gap ⅛
The gap is multiplied by the ratio every single step. That is the mechanism, and it says two things at once: the gap never reaches zero, so the running total never actually arrives; and the gap never stops shrinking, so it gets closer than any distance you might name. The total is not a number the adding reaches. It is the number the adding gets arbitrarily close to and never passes.
With a negative ratio the same thing happens from alternate sides — the running total overshoots, then undershoots, by less each time.
Turning a Repeating Decimal into a Fraction
Here is the everyday use, and it is the reason every repeating decimal is a rational number.
0.272727… is 0.27 + 0.0027 + 0.000027 + … That is geometric: first term 27/100, ratio 1/100. So the total is (27/100) ÷ (99/100) = 27/99 = 3/11. Type those two numbers into the boxes above and you get 3/11 exactly, not 0.2727 rounded.
The recipe: the repeating block over 10 to the power of its length is the first term, and 1 over 10 to that same power is the ratio. Two digits repeating gives a ratio of 1/100; three gives 1/1000.
And the one that starts arguments: 0.999… First term 9/10, ratio 1/10, so the sum is (9/10) ÷ (9/10) = 1. Not nearly 1. It is a second way of writing the number 1, exactly as 2/4 is a second way of writing ½.
How to Use This Infinite Geometric Series Calculator
Three boxes, and you fill in any two. There is no count here — the series runs forever, so the only questions are whether it settles and where.
a₁ and r gives the total. Leave a₁ blank instead and it works backwards from the total. Leave r blank and it finds the ratio — which is worth noticing, because r = 1 − a₁/S comes out cleanly here, while in a geometric series calculator for a fixed number of terms the ratio is trapped inside r to the power n and no rearranging frees it. Running forever is what sets it loose.
| first term a₁ | Where the series starts |
| common ratio r | What each term is multiplied by. Must be strictly between −1 and 1 for a total to exist |
| the sum S | The total. Give it to work backwards, or leave it blank |
| ← → ↑ ↓ | Move between the three boxes |
| ± | Flips the sign of the box you are in |
| ⁄ and . | Fraction bar and decimal point. 3⁄4 stays a fraction all the way through |
| ⌫ / AC | Delete one character, or clear all three boxes |
| = | Works it out |
| My keyboard | Raises your own keyboard instead of the pad, for that box |
The total is always exact. The running totals underneath switch to rounded decimals when their exact fractions would run to twenty digits — a ratio of 1/66 does that by the seventh term — because a fraction nobody can read hides the very shrinking it is there to show. When that happens the page says so, and the answer itself is never rounded.
Infinite Geometric Series FAQ
When does an infinite geometric series have a sum?
What is the formula for the sum of an infinite geometric series?
What happens when the ratio is exactly -1?
Does 0.999... really equal 1?
How do you turn a repeating decimal into a fraction?
Can you find the ratio if you know the total?
Check the ratio, then divide. Everything else on this page exists to make the strange part believable — that a sum with no last term still has an answer, and that you can watch it arrive.
