Sum of Series Calculator

Background.

This calculator evaluates a summation written in sigma notation: pick a general term, set the index bounds, and get the total. It handles three families of general term — linear (Ak + B), power (k, k² or k³), and geometric (A × rᵏ) — and it sums between any two whole-number bounds you like, not just from the beginning.

That last point is the reason this page exists alongside the arithmetic sequence and geometric sequence calculators. Those two pages also report a sum, but always the sum of the first n terms, starting from the first. Neither can answer 'what is the sum of k² from k = 3 to k = 50', because k² is not an arithmetic or a geometric term at all, and neither can start the index anywhere except the beginning. If what you actually want is the nth term of a progression, or whether an infinite geometric series converges, those pages are the right ones and this page links to them — the overlap is real and it is stated rather than hidden.

Every result here comes from a closed form, not from a loop. The three power sums are the ones OpenStax's Calculus Volume 1 publishes as the rule 'Sums and Powers of Integers': the sum of the first n integers is n(n+1)/2, the sum of their squares is n(n+1)(2n+1)/6, and the sum of their cubes is n²(n+1)²/4. Summing between arbitrary bounds then costs one extra subtraction, since the total from m to n is simply the total from 1 to n minus the total from 1 to m − 1. Because none of this involves iterating, you can set the upper bound to ten million and get the answer instantly and exactly.

The exponent stops at 3, and the reason is worth saying plainly rather than burying: the closed forms for k, k² and k³ are the three published in the source this page cites. The sum of fourth powers needs Faulhaber's formula and Bernoulli numbers, and rather than typing a fourth polynomial from memory, the page narrows its scope and the error message tells you why. A wrong polynomial would be indistinguishable from a right one until someone checked it.

Geometric mode uses the finite geometric series formula from OpenStax College Algebra 2e, adjusted for a lower bound that need not be zero. That formula divides by 1 − r, so r = 1 is handled as its own case — with every term equal, the sum is just A multiplied by the number of terms. A ratio of exactly zero is refused, both because every term past the first would vanish and because 0⁰ at a lower bound of zero has no agreed value. Geometric sums are also checked for size before anything is computed: a ratio of 10 over a thousand terms produces a number with a thousand digits, and the page refuses up front instead of grinding.

One convention to note before reading any result: both bounds are inclusive. Summing from k = 0 to k = 10 adds eleven terms, not ten, and the number-of-terms output is shown beside the total so you can confirm the calculator is summing what you intended.

What is sum of series calculator?

Sigma notation is a compact way of writing a sum: the capital Greek sigma, an index variable with a starting value below it and an ending value above it, and a general term to the right that depends on the index. Σ from k = 1 to 10 of (3k + 2) means 'substitute k = 1, 2, 3, … 10 into 3k + 2 and add the results'. A series is what you get when you add the terms of a sequence, so every summation on this page is a series, and the number it produces is the series sum. Some series have closed forms — compact formulas that give the total without adding anything — and this calculator uses those exclusively. OpenStax Calculus Volume 1 §5.1 publishes the three power-sum formulas under the rule 'Sums and Powers of Integers', and OpenStax College Algebra 2e §9.4 gives the finite geometric series formula Sn = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. Closed forms are what make summing ten million terms as fast as summing ten.

How to use this calculator.

1. 01Choose the shape of your general term: linear (Ak + B), a power of k, or geometric (A × rᵏ).
2. 02Fill in the coefficients that mode uses — the others are ignored, so leave whatever is in them.
3. 03Set the lower and upper bounds of the index k. Both are inclusive.
4. 04Read the sum, then check the sigma-notation line beneath it to confirm the calculator is summing the expression you meant.
5. 05Use the first-term and last-term outputs as a quick sanity check on your coefficients before trusting the total.

The formula.

Every mode reduces to a closed form and a subtraction. Write F(N) for the total from index 1 up to N; then the total from m to n is F(n) − F(m − 1), because subtracting the head of the sum leaves exactly the piece you asked for. For power sums, F is one of the three polynomials OpenStax Calculus Volume 1 §5.1 publishes: N(N+1)/2 for k, N(N+1)(2N+1)/6 for k², and N²(N+1)²/4 for k³. Those polynomials evaluate to zero at N = 0 and at N = −1, which is exactly what an empty sum should give — and it is why a lower bound of 0 needs no special handling here. For a linear general term Ak + B, the sum splits into A times the sum of the indices plus B added once per term: A·(F₁(n) − F₁(m−1)) + B·(n − m + 1). For a geometric general term A·rᵏ, factoring rᵐ out of the front turns the sum into a standard finite geometric series, giving A·rᵐ·(r^(n−m+1) − 1)/(r − 1), which reduces to the familiar formula in OpenStax College Algebra 2e §9.4 when m = 0. That expression divides by r − 1, so r = 1 is computed separately as A multiplied by the number of terms rather than being allowed to become 0/0. All arithmetic runs at full decimal precision and is rounded once, at the end, to 10 decimal places — nothing is rounded at an intermediate step. Geometric sums get one extra safeguard: the size of the result is predicted in base-10 logarithms before any power is computed, so a request like 10ᵏ summed to k = 1000 is refused immediately rather than building a thousand-digit number and stalling the page.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax (Rice University), Calculus Volume 1, §5.1 'Approximating Areas', rule 'Sums and Powers of Integers' — Σi = n(n+1)/2, Σi² = n(n+1)(2n+1)/6, Σi³ = n²(n+1)²/4. These three closed forms are the whole basis of the power mode and the reason its exponent stops at 3. Retrieved 2026-07-29; open access.
2. [2]OpenStax (Rice University), College Algebra 2e, §9.4 'Series and Their Notations' — the finite geometric series Sn = a₁(1 − rⁿ)/(1 − r) for r ≠ 1, and the arithmetic series Sn = n(a₁ + aₙ)/2 used as the cross-check on linear mode. Retrieved 2026-07-29; open access.
3. [3]Encyclopedia of Mathematics (European Mathematical Society), 'Geometric progression' — independent check on the geometric closed form, Sn = a₀(1 − qⁿ)/(1 − q), together with the requirement that the ratio q be non-zero. Both sources give 2047 for the sum of 2ᵏ from k = 0 to 10, matching the doubling identity 2ⁿ⁺¹ − 1. Retrieved 2026-07-29; open access.
4. [4]OpenStax (Rice University), College Algebra 2e, §9.2 'Arithmetic Sequences' — aₙ = a₁ + d(n − 1), cited for the relationship between this page's linear mode at a lower bound of 1 and the arithmetic sequence calculator's sum output. Retrieved 2026-07-29; open access.