Power Sum Calculator

Power (k):

Number of terms (n):

Calculation Steps

Power Sum Calculator Introduction

The power sum calculator computes the sum of the k-th powers from 1 to n, using the formula below:

S_k(n) = 1^k + 2^k + 3^k + ⋯ + n^k

Power (k)	Name	Sum Formula
0	Zeroth Power Sum	S₀(n) = n
1	Natural Number Sum (First Power Sum)	S₁(n) = ⁿ⁽ⁿ⁺¹⁾⁄₂
2	Square Sum (Second Power Sum)	S₂(n) = ^n(n+1)(2n+1)⁄₆
3	Cube Sum (Third Power Sum)	S₃(n) = [ⁿ⁽ⁿ⁺¹⁾⁄₂]²
4	Fourth Power Sum	S₄(n) = ^{n(n+1)(2n+1)(3n²+3n-1)}⁄₃₀
5	Fifth Power Sum	S₅(n) = ^{n²(n+1)²(2n²+2n-1)}⁄₁₂

General Formula (Using Bernoulli Numbers):

S_k(n) = ¹⁄_k+1 ∑_j=0^k C(k+1,j) B_j n^k+1-j

Here, B_j denotes the Bernoulli numbers, and C(k+1,j) denotes the binomial coefficient.

(k+1)S_k(n) = (n+1)^k+1 - 1 - ∑_i=0^k-1 C(k+1,i) S_i(n)

This recursive formula can be used to compute higher-order power sums.