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Power Sum — Definition, Formula & Examples

A power sum is the sum of the kkth powers of the first nn positive integers, written as 1k+2k+3k+⋯+nk1^k + 2^k + 3^k + \cdots + n^k. Familiar cases include the sum of integers (k=1k=1), the sum of squares (k=2k=2), and the sum of cubes (k=3k=3).

For non-negative integers nn and kk, the power sum Sk(n)S_k(n) is defined as Sk(n)=∑i=1nikS_k(n) = \sum_{i=1}^{n} i^k. Each power sum is a polynomial of degree k+1k+1 in nn, and closed-form expressions exist for every positive integer kk, recoverable via Faulhaber's formulas or Bernoulli numbers.

Key Formula

Sk(n)=∑i=1nikS_k(n) = \sum_{i=1}^{n} i^k
Where:
  • nn = The number of terms (a positive integer)
  • kk = The exponent applied to each term (a non-negative integer)
  • Sk(n)S_k(n) = The resulting power sum

How It Works

To evaluate a power sum, you apply the known closed-form formula for the specific exponent kk. For k=1k=1, the result is n(n+1)2\frac{n(n+1)}{2}. For k=2k=2, it is n(n+1)(2n+1)6\frac{n(n+1)(2n+1)}{6}. For k=3k=3, it simplifies to [n(n+1)2]2\left[\frac{n(n+1)}{2}\right]^2, which means the sum of cubes equals the square of the sum of integers. Higher-order power sums can be derived recursively using Bernoulli numbers or Pascal's triangle identities.

Worked Example

Problem: Find the sum of the squares of the first 10 positive integers: 12+22+⋯+1021^2 + 2^2 + \cdots + 10^2.
Identify the formula: For k=2k = 2, the closed-form is:
S2(n)=n(n+1)(2n+1)6S_2(n) = \frac{n(n+1)(2n+1)}{6}
Substitute $n = 10$: Plug in n=10n = 10:
S2(10)=10⋅11⋅216=23106=385S_2(10) = \frac{10 \cdot 11 \cdot 21}{6} = \frac{2310}{6} = 385
Answer: 12+22+⋯+102=3851^2 + 2^2 + \cdots + 10^2 = 385

Why It Matters

Power sums appear throughout discrete mathematics and physics whenever you need to total quantities that grow polynomially. They are essential in deriving big-O complexity for nested loops in computer science and in analytic number theory results like Waring's problem, which asks how many kkth powers are needed to represent any positive integer.

Common Mistakes

Mistake: Confusing the sum-of-cubes formula with the cube of the sum-of-integers formula.
Correction: The sum of cubes ∑i3=[n(n+1)2]2\sum i^3 = \left[\frac{n(n+1)}{2}\right]^2 equals the *square* of the sum of integers, not the cube. Keep the exponent straight: it is (S1(n))2\left(S_1(n)\right)^2, not (S1(n))3\left(S_1(n)\right)^3.