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alternate form of sum of rth powers of the first n positive integers


We will show that

nk=0kr=n+11br(x)𝑑x

We need two basic facts. First, a property of the Bernoulli polynomialsMathworldPlanetmathPlanetmath is that br(x)=rbr-1(x). Second, the Bernoulli polynomials can be written as

br(x)=rk=1(rk)Br-kxk+Br

We then have

n+11br(x) =1r+1(br+1(n+1)-br+1(1))=1r+1r+1k=0(r+1k)Br+1-k((n+1)k-1)
=1r+1r+1k=1(r+1k)Br+1-k(n+1)k

Now reverse the order of summation (i.e. replace k by r+1-k) to get

n+11br(x)=1r+1rk=0(r+1r+1-k)Bk(n+1)r+1-k=1r+1rk=0(r+1k)Br(n+1)r+1-k

which is equal to nk=0kr (see the parent (http://planetmath.org/SumOfKthPowersOfTheFirstNPositiveIntegers) article).

Title alternate form of sum of rth powers of the first n positive integers
Canonical name AlternateFormOfSumOfRthPowersOfTheFirstNPositiveIntegers
Date of creation 2013-03-22 17:46:10
Last modified on 2013-03-22 17:46:10
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Proof
Classification msc 11B68
Classification msc 05A15