Series Sum Calculator

Series Sum Formulas: From Gauss to Faulhaber

Few mathematical discoveries delight students more than the realization that adding long lists of numbers can be done instantly with a single formula. The story of series sums stretches from ancient Greece to 17th-century Germany, and the underlying ideas power everything from algorithm complexity analysis to physics simulations.

The Gauss Legend: 1 + 2 + ... + 100 = 5050

Around 1787, a young Carl Friedrich Gauss was reportedly asked by his teacher to add all integers from 1 to 100 — a task meant to keep the class busy. Within seconds, Gauss wrote 5050. His insight: pair the first and last terms (1+100=101), the second and second-to-last (2+99=101), and so on, giving 50 pairs each summing to 101. The result is 50 × 101 = 5050. This elegant pairing argument generalizes to the famous triangular number formula:

S(n) = n(n+1)/2     [Sum of first n positive integers]

This is also written as the n-th triangular number Tₙ, because n(n+1)/2 dots can always be arranged in a perfect equilateral triangle.

Sum of Squares and Cubes

The sum of squares formula was known to ancient Greek and Chinese mathematicians and appears in combinatorics, statistics, and physics:

Σk² = n(n+1)(2n+1)/6     [Sum of squares, 1² + 2² + ... + n²]

The sum of cubes obeys one of mathematics' most surprising identities — Nicomachus's theorem, proven around 100 AD:

Σk³ = [n(n+1)/2]² = (Σk)²     [Sum of cubes = square of sum!]

This means 1³ + 2³ + 3³ + 4³ + 5³ = 225 = 15² = (1+2+3+4+5)². Every time you add the first n cubes, you get a perfect square.

Faulhaber's Formula and Higher Powers

Johann Faulhaber (1580–1635), a German mathematician and reckoning master, computed sums of powers Σk¹ through Σk¹⁷ by hand — an extraordinary achievement. Today we know these as Faulhaber's formulas. Each sum Σkᴸ is a polynomial of degree p+1 in n, with the leading coefficient 1/(p+1):

p	Formula for Σk=1n kᴸ
1	n(n+1)/2
2	n(n+1)(2n+1)/6
3	[n(n+1)/2]²
4	n(n+1)(2n+1)(3n²+3n−1)/30
5	n²(n+1)²(2n²+2n−1)/12
6	n(n+1)(2n+1)(3n⁴+6n³−3n+1)/42

Arithmetic and Geometric Series

An arithmetic series adds terms that increase by a fixed common difference d. With first term a and n terms, the sum is:

S = n/2 × (2a + (n−1)d) = n × (a + last term) / 2

A geometric series multiplies each term by a fixed common ratio r. With first term a and n terms:

S = a(r⊃n; − 1) / (r − 1)    when r ≠ 1   ;    S = n × a    when r = 1

Geometric series appear in compound interest (your investment grows geometrically), repeating decimals (0.333... = 1/3 as an infinite geometric series with r = 1/10), and signal processing.

Applications in Algorithm Complexity and Beyond

• Algorithm analysis: Selection sort makes 1+2+...+(n-1) = n(n-1)/2 comparisons, giving O(n²) complexity — directly from the sum of integers formula.
• Physics: The total distance fallen under gravity after n seconds involves Σk (arithmetic series); moments of inertia involve Σk².
• Finance: Compound interest over n periods is a geometric series; annuity payments sum to a closed-form geometric formula.
• Statistics: The variance of a uniform distribution over {1, ..., n} involves Σk² and Σk.
• Computer graphics: Bezier curves and B-splines use polynomial bases whose integrals are related to power sums.

Reference: Sum Values for Small n

n	Σk	Σk²	Σk³	Σk⁴
1	1	1	1	1
2	3	5	9	17
3	6	14	36	98
5	15	55	225	979
10	55	385	3025	25333
20	210	2870	44100	722666
100	5050	338350	25502500	2050333330