Infinite geometric series sum
Enter the first term and the common ratio to get the total of an infinite geometric series: a first term of 1 with a ratio of 0.5 adds up to 2. Ratios of absolute value 1 or more are refused instead of turning into infinity. Free in your browser.
- Free
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- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 1
- Resistance (Ω)
- 0.5
- Sum
- 2
An infinite geometric series adds forever: start with a first term a, multiply by a fixed common ratio r at each step, and keep going without end. When the absolute value of that ratio is strictly less than one, the partial sums settle on a single finite number given by the closed form S equals a divided by one minus r. This calculator takes a and r, checks the convergence condition, and returns the exact floating-point sum so students, teachers, and engineers do not have to re-derive the formula under exam pressure or in a spreadsheet. Ratios with absolute value one or greater are rejected with a clear validation error instead of a silent infinity or NaN.
How the infinite geometric sum is defined
A geometric series has each term equal to the previous term times a constant ratio r. Written out, the infinite sum is a plus a times r plus a times r squared plus a times r cubed and so on without a last term. When the absolute value of r is strictly less than one, the remaining tail shrinks toward zero and the total converges to the closed form S equals a over one minus r.
That identity is the only arithmetic this service performs after validating inputs: no iterative accumulation of thousands of terms, no floating-point drift from long loops, and no approximation tables. Classic classroom checks include a equals one and r equals one half, which sums to two; a equals ten and r equals one tenth, which sums to about eleven point one one one; and a equals three with r equals negative one half, which alternates and still converges to two.
The same formula covers every finite a as long as the ratio condition holds.
Where infinite geometric sums show up in real work
Secondary-school and college algebra courses constantly ask for the sum of an infinite geometric series given a and r; a trusted calculator with a strict convergence check speeds homework grading and removes spreadsheet mistakes where someone divides by one minus r even when the series diverges. Engineering and finance teaching materials use geometric series for present-value intuition, repeated interest factors, and infinite geometric decay of error terms—having a shared numeric oracle keeps lecture demos and student code aligned.
Game and animation tutorials often model shrinking bounces or fade factors as geometric sequences; the infinite sum answers how much total displacement or intensity remains if the pattern continues forever. If you later need the partial sum of the first n terms, term-by-term expansion, or complex ratios, treat those as separate search intents rather than overloading this infinite-sum landing.
What you can do with it
Homework and exam checks
Verify S = a/(1−r) for classroom problems and reject divergent |r| ≥ 1 cases with a clear error instead of Infinity.
Teaching convergence with live numbers
Show students how halving ratios converge while |r| ≥ 1 fails, using the same function in browser demos and grading scripts.
Engineering and finance teaching demos
Compute infinite geometric totals for decay, present-value intuition, and series benchmarks without re-deriving the closed form each time.
FAQ
What formula does this calculator use?
S = a / (1 − r) for an infinite geometric series that starts at a and multiplies by r each step, provided |r| < 1.
What happens if |r| is 1 or greater?
The infinite series does not converge to a finite sum when the absolute ratio is at least one.
Can a or r be negative?
Yes. Negative first terms and negative ratios are allowed as long as |r| is strictly less than 1 (for example r = −0.5 converges).
Does this return the sum of only the first n terms?
Finite partial sums of n terms are a different intent and formula.
What does it cost?
The browser tool is free and runs locally.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/math/geometric-series-sum \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":1,"r":0.5}'const res = await fetch("https://api.kit.forhosting.com/math/geometric-series-sum", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 1,
"r": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/geometric-series-sum",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 1,
"r": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/geometric-series-sum", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":1,"r":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":1,"r":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/geometric-series-sum", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 1,
"r": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.geometric_series_sum",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
The free browser tool and the paid API share one pure function, so classroom demos and automated pipelines never silently diverge on the same inputs.
How the infinite geometric sum is defined
If |r| is greater than or equal to one, the terms do not die out—the series diverges—and this endpoint refuses the request rather than inventing a finite answer. That hard gate matches how textbooks state the theorem and keeps downstream code from treating Infinity as a normal sum.
What you send and what you get back
After parsing, the service checks that the absolute value of r is strictly less than one; if |r| is at least one, the error message states that the absolute ratio must be strictly less than one so form clients and scripts can correct the input without guessing. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts when you need bulk evaluation of geometric totals without reimplementing the gate in every language. Provide two required finite numbers: a (aliases first, a1, first_term) for the first term, and r (aliases ratio, common_ratio) for the common ratio. Missing fields, non-numeric strings, NaN, and Infinity are rejected with invalid_input and are not charged. On success the JSON response echoes the normalized a and r, returns sum as a over one minus r, sets converges to true, and includes formula as the string a/(1-r) for documentation in logs and teaching UIs. Example checks: a equals one and r equals zero point five yields sum two; a equals eight and r equals zero point two five yields sum about ten point six six seven; r equals one or r equals negative one or r equals two all fail invalid_input.
Where infinite geometric sums show up in real work
Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.