Consider the sum of the m-th powers of the first n integers:

Suppose that the sum is a polynomial in *n*: . Show that is an *m*+1 degree polynomial with a leading coefficient of .

Consider the integral . We know that the value of this integral is

.

However, we can of course express the integral as the limit of a Riemann Sum (Riemann Integral). If we divide the interval into *n* equally sized partitions, and choose the right side of the partition for the height, then the *k*-th partition has width and height . Then the Riemann Sum is:

But the limit of this sum as is the integral, and so:

. For a polynomial in *n*, this means the highest power term in n must be , and so is an *m*+1 degree polynomial with a leading coefficient of .

We can also see from the *n*=0 case that and so the constant term in must be zero. Further, for *n*=1, we have and so the sum of the coefficients of must be unity.

The first few of these polynomials are known to many:

.

These polynomials can be found by Faulhaber’s Formula, which gives them in terms of the Bernoulli numbers (or related Bernoulli polynomials).

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