Power Sum Calculator

About the Sum of Powers Calculator

The sum of powers calculator computes the sum of the k-th powers of integers from 1 to n, given by: Sk(n) = 1ᵏ + 2ᵏ + 3ᵏ + ⋯ + nᵏ.

Common Sum of Powers Formulas

• Power (k) = 0, sum of zeros: S0(n) = n
• k = 1, sum of natural numbers: S1(n) = n(n+1)/2
• k = 2, sum of squares: S2(n) = n(n+1)(2n+1)/6
• k = 3, sum of cubes: S3(n) = [n(n+1)/2]²
• k = 4, sum of fourth powers: S4(n) = n(n+1)(2n+1)(3n²+3n-1)/30
• k = 5, sum of fifth powers: S5(n) = n²(n+1)²(2n²+2n-1)/12

General Sum Formula

General formula (using Bernoulli numbers): Sk(n) = 1/(k+1) Σⱼ₌₀ᵏ C(k+1, j) Bⱼ n^(k+1-j), where Bⱼ are the Bernoulli numbers and C(k+1, j) is a binomial coefficient.

Recurrence relation: (k+1)Sk(n) = (n+1)^(k+1) - 1 - Σᵢ₌₀ᵏ⁻¹ C(k+1, i) Si(n). This recurrence can be used to compute higher power sums.

Bernoulli Numbers (First Few)

• n=0: B0 = 1
• n=1: B1 = -1/2
• n=2: B2 = 1/6
• n=4: B4 = -1/30
• n=6: B6 = 1/42
• n=8: B8 = -1/30
• n=10: B10 = 5/66