Sum of Series Calculator

Background.

Adding the squares from 3² up to 50² by hand means 48 multiplications and 47 additions. With the right formula it is one line: the first 50 squares total 50 × 51 × 101 ÷ 6 = 42,925, and taking away 1² + 2² = 5 leaves 42,920. That move, summing everything from the start and then subtracting the part before your first index, is how every total here is found, so an upper bound of ten million takes no longer than an upper bound of ten.

Three shapes of term are covered. A linear term, Ak + B, gives an arithmetic series such as 5 + 8 + 11 + … + 32. A power term, k, k² or k³, gives the sums of integers, squares and cubes that turn up in area approximations in calculus, in counting puzzles and in the sums of squares behind a variance. A geometric term, A × rᵏ, gives a series in which each term is a fixed multiple of the one before: doubling, halving, or flipping sign when r is negative.

The index may start anywhere from 0 upward, which is what separates a series total from the sum of the first n terms that a sequence calculator reports. Both bounds count, so k = 0 to 10 is eleven terms. Reciprocal sums such as 1 + 1/2 + 1/3 + …, and infinite geometric series, are left to their own calculators.

What is sum of series calculator?

A series is the total you get by adding the terms of a sequence, and sigma notation is its shorthand: a capital Σ with the starting index written beneath, the final index above, and the general term to the right. Σ from k = 1 to 10 of (3k + 2) means substitute k = 1, 2, …, 10 into 3k + 2 and add the ten results. A finite series always has a value; the useful question is whether it has a closed form, a formula that gives the total without adding term by term. Linear terms, powers up to cubes and geometric terms all have one, and those are the three families offered.

How to use this calculator.

1. 01Under "What are you summing?", pick the shape of the term inside the sigma: linear (Ak + B), power (kᵖ) or geometric (A × rᵏ).
2. 02Fill in the fields your choice uses: A and B for linear, the exponent p (1, 2 or 3) for power, A and the ratio r for geometric. The others are ignored, so they can stay as they are. In geometric mode neither A nor r may be 0, and a negative r makes the terms alternate in sign.
3. 03Set the lower and upper bounds as whole numbers from 0 to 10,000,000, with the upper bound no smaller than the lower. Both ends are included.
4. 04Before using the total, compare the sigma-notation line and the first and last terms with what you meant. A first term that looks wrong usually means the index should start at 1 rather than 0, or the reverse.

The formula.

Write F(N) for the sum from k = 1 to N. The sum from m to n is then F(n) − F(m − 1): everything up to n, minus the stretch before m. For power terms F is one of the three classical closed forms that OpenStax Calculus Volume 1 lists as sums and powers of integers: Σk = N(N + 1)/2, Σk² = N(N + 1)(2N + 1)/6 and Σk³ = N²(N + 1)²/4. Each of these polynomials equals 0 at N = 0 and at N = −1, the value an empty sum should have, so a lower bound of 0 needs no special case. Fourth and higher powers need Faulhaber's formula, which brings in Bernoulli numbers, and are not offered.

A linear term splits into A times the sum of the indices plus B once for each term: A[F₁(n) − F₁(m − 1)] + B(n − m + 1), where F₁ is the integer-sum formula. A geometric term factors as A·rᵐ(1 + r + r² + … + r^(n−m)), and the bracket is the finite geometric series from OpenStax College Algebra, so the total is A·rᵐ(r^(n−m+1) − 1)/(r − 1). That expression divides by r − 1, so when r = 1, and every term is simply A, the total is computed as A multiplied by the number of terms.

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.