Sum of Powers Calculator

Calculate $1^k+2^k+3^k+\cdots+n^k$ with formulas, exact results, and step-by-step explanations.

What Is a Sum of Powers?

A sum of powers adds consecutive positive integers after raising each integer to the same exponent. In sigma notation, the general form is:

$$ S_k(n) = \sum_{i=1}^{n}i^k = 1^k+2^k+3^k+\cdots+n^k $$

The value of $k$ determines the type of powers being added. For example, $k=1$ produces the sum of integers, $k=2$ produces the sum of squares, and $k=3$ produces the sum of cubes.

This calculator provides direct formulas for the first several powers and calculates higher powers term by term.

Sum of Powers Formulas

Power $k=0$

Sum of Zeroth Powers

$$ \sum_{i=1}^{n}i^0=n $$

Every non-zero number raised to the zeroth power equals 1, so there are $n$ terms whose total is $n$.

Power $k=1$

Sum of First Powers

$$ \sum_{i=1}^{n}i = \frac{n(n+1)}{2} $$

This is the familiar formula for the sum of the first $n$ positive integers.

Power $k=2$

Sum of Squares

$$ \sum_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6} $$

The formula gives the sum $1^2+2^2+\cdots+n^2$.

Power $k=3$

Sum of Cubes

$$ \sum_{i=1}^{n}i^3 = \left( \frac{n(n+1)}{2} \right)^2 $$

The sum of the first $n$ cubes is the square of the sum of the first $n$ positive integers.

Power $k=4$

Sum of Fourth Powers

$$ \sum_{i=1}^{n}i^4 = \frac{ n(n+1)(2n+1)(3n^2+3n-1) }{30} $$

This formula calculates the sum $1^4+2^4+\cdots+n^4$.

Examples of Sums of Powers

Example 1: $k=1$

$$ 1+2+3+4+5=15 $$

Using the formula: \[ \frac{5(6)}{2}=15. \]

Example 2: $k=2$

$$ 1^2+2^2+3^2+4^2+5^2 = 55 $$

Using the square-sum formula gives $55$.

Example 3: $k=3$

$$ 1^3+2^3+3^3+4^3+5^3 = 225 $$

This also equals: \[ (1+2+3+4+5)^2=15^2=225. \]

Sum of Powers and Sigma Notation

The Greek letter sigma, $\Sigma$, is commonly used to represent a summation. Instead of writing every term separately, we can write:

$$ \sum_{i=1}^{n}i^k $$

The lower limit $i=1$ tells us where the summation starts, while $n$ is the upper limit. The exponent $k$ specifies the power applied to each integer.

For example:

$$ \sum_{i=1}^{5}i^3 = 1^3+2^3+3^3+4^3+5^3 = 225 $$

Higher Powers and Faulhaber's Formula

For every fixed non-negative integer $k$, the sum \[ 1^k+2^k+\cdots+n^k \] can be expressed as a polynomial in $n$. These formulas are commonly associated with Faulhaber's formula.

$$ \sum_{i=1}^{n}i^k $$

The polynomial becomes more complicated as $k$ increases. For this reason, the calculator uses the well-known closed forms for powers 0 through 4 and performs direct exact summation for higher powers.

For large values of $n$ and higher powers, direct calculation can require substantially more computation than a closed-form formula.

How to Use the Sum of Powers Calculator

Enter the upper limit $n$, such as $10$.
Enter the exponent $k$, such as $2$ or $3$.
Click Calculate Sum of Powers.
The calculator displays the mathematical expression and exact result.
For supported low powers, the page also shows the formula and calculation steps.

For example, entering $n=10$ and $k=2$ calculates: \[ 1^2+2^2+\cdots+10^2=385. \]

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Sum of Powers FAQ

What is the sum of powers formula?

The general sum of powers is written as \[ \sum_{i=1}^{n}i^k. \] For each fixed non-negative integer $k$, this sum can be represented by a polynomial in $n$.

What is the formula for $1+2+\cdots+n$?

The formula is \[ \frac{n(n+1)}{2}. \] It is the sum of the first $n$ positive integers.

What is the formula for the sum of squares?

The sum of the first $n$ squares is \[ \sum_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6}. \]

What is the formula for the sum of cubes?

The sum of the first $n$ cubes is \[ \sum_{i=1}^{n}i^3 = \left( \frac{n(n+1)}{2} \right)^2. \]

What is the sum of fourth powers?

The sum of the first $n$ fourth powers is \[ \sum_{i=1}^{n}i^4 = \frac{ n(n+1)(2n+1)(3n^2+3n-1) }{30}. \]

What does $k$ represent in a sum of powers?

The variable $k$ is the exponent applied to each integer. For example, $k=2$ means squares, while $k=3$ means cubes.

Can this calculator calculate powers greater than 4?

Yes. For $k>4$, the calculator evaluates the terms $1^k,2^k,\ldots,n^k$ directly and adds them using exact integer arithmetic when BCMath is available.