Series Summation Calculator
Sum series in closed form: Σ of a polynomial term with the standard sums, the nth term and sum from a sequence's first terms by the method of differences, the Vₙ (telescoping) method, arithmetico-geometric and harmonic progressions.
About the Series Summation Calculator
Σk = n(n + 1)/2, Σk² = n(n + 1)(2n + 1)/6 and Σk³ = [n(n + 1)/2]² turn the sum of any polynomial term into a polynomial in n. When only the first terms are known, the method of differences finds the rule: if the kth differences are constant, the nth term is a polynomial of degree k.
A sum like 1/(1·2·3) + 1/(2·3·4) + … telescopes (the Vₙ method): each term is the difference of two simpler ones, so almost everything cancels. In an AGP each term is an AP term times a GP term; an HP's reciprocals form an AP.
The formula
Σk² = n(n + 1)(2n + 1)/6; AGP: S∞ = a/(1 − r) + dr/(1 − r)²
Worked example
3, 8, 15, 24, 35, …
- Differences 5, 7, 9, 11; second differences 2 (constant)
- Tₙ = n(n + 2), Sₙ = n(n + 1)(2n + 7)/6
Questions people ask
Does the method of differences always work?
It finds the polynomial rule when one exists. Some sequences (like 1, 2, 4, 8) have differences that never become constant; those are geometric.
Is there a formula for the sum of an HP?
No simple one; the page gives the nth term and the harmonic mean.