How To Find The Sum Of A Geometric Series Calculator
In a geometric series each term is the previous one multiplied by a fixed ratio. That single change turns polynomial growth into exponential growth — and turns the sum formula into something with a constraint worth respecting.
Sum of the first n terms
5.9941
- Last term (aₙ)
- 0.0059
- Sum to infinity — 0 means it diverges
- 6
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
The formula this calculator uses
S = a1 (1 - r^n) / (1 - r), and S-infinity = a1 / (1 - r) when |r| < 1
- a1
- First term of the series
- r
- Common ratio between consecutive terms
- n
- Number of terms being added
- |r|
- Absolute size of the ratio — below 1 means the terms shrink
- S-infinity
- Limit the partial sums approach, when one exists
How to check the result by hand
- 1
Verify the ratio is constant
Divide three consecutive pairs. If the quotients differ, the series is not geometric — the likely alternative is arithmetic, where the differences rather than the ratios are constant.
- 2
Read off a1, r and n
The ratio is term-after divided by term-before, not the other way round. For a quantity growing 8 percent a year, r is 1.08, because the new value is 108 percent of the old.
- 3
Check whether |r| is below one
This decides which questions have answers. Below one, the terms shrink and an infinite limit exists. At or above one, there is no ceiling and quoting S-infinity is meaningless.
- 4
Apply the finite formula
Compute r^n first, then 1 - r^n, then scale by a1 / (1 - r). Here that is 0.5^10 = 0.0009765625, giving 3 x 0.9990234375 / 0.5 = 5.994140625.
- 5
Compare against the ceiling when one exists
The infinite limit is 3 / 0.5 = 6. The gap of 0.005859375 equals the next term, which is always true for a geometric series and is a quick way to catch a mistyped ratio.
For worked examples, common mistakes and the limits of this formula, read the full How To Find The Sum Of A Geometric Series page.