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Ken Ward's Mathematics Pages

Series Sums of the Powers of the First n Natural Numbers - General


  1. General Formula Using Summation
  2. Examples of Powers of the Natural Numbers

General Formula Using Summation

In previous pages we have looked at various ways to sum the powers of the natural numbers: powers of 1 and 2. Here we will generalise and find one (of many) formulae to give us the sums of powers with much less work.

Some of the techniques we examined, worked only for some of the powers. The technique of summation works for all powers.

In general, the sum of the (k+1) terms is the sum of the k terms plus  the (k+1) term:
powers2.gif [1.1]

Expanding the left-hand side, using the Binomial Theorem, we get:
powers3.gif [1.2]

Replacing the left-hand side of 1.1 with the right-hand side of 1.2, we get:

powers4.gif [1,3]

As expected the km+1 terms will cancel.

Leaving the km sum where it is, we can move the rest to the right-hand side;
powers5.gif [1.4]

More generally, we can say:
powers6.gif [1.5]
This is a recursive formula, of course, so you can find the sum of the m powers only when you know the sum of the (m-1) powers. By setting m=0, 1, 2, 3...m, you can find the sum of any m power. .

Let us use the formula, by setting m=0.

Because the lower sum, r=2 exceeds the upper 1, then all that remains is (n+1) which is the sum of 1's from 0 to n. ∑n01=(n+1)!

Let m=3, to find the sum of the first n cubes. Substituting in the equation:

We find:

Giving us the formula:

The sum of the cubes of the first n natural numbers is the square of the formula for the first n natural numbers, so it is easy to remember!

Examples of Powers of the Natural Numbers

The following table collects information about the coefficients of the sums of the first n powers of the natural numbers.
Power m+1 m m-1 m-2 m-3 m-4 m-5
1 1/2 1/2
2 1/3 1/2 1/6 sumSquares6.gif
3 1/4
4 1/5
-1/30 sum4thFormula.gif
5 1/6 1/2 5/12 -1/12 sum5thFormula.gif
6 1/7 1/2 1/2 -1/6 1/42 sum6thFormula.gif
7 1/2 7/12 -7/24 1/12 sum7thPower.gif

Ken Ward's Mathematics Pages

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