sum of th powers of the first positive integers
A very classic problem is this: what is the sum of the numbers from to ? The answer is , and there are a number of ways to see it. Before stating them, we generalize the problem somewhat: What is ?
Write the sum twice:
which is exactly , so
Having seen this, it is natural to generalize this question further, and ask: What is , for any integer ?
This question, being so general, does not have such a tidy answer, but it can be solved.
Observe that this formula is a polynomial in of degree (http://planetmath.org/OrderAndDegreeOfPolynomial) ; for a given , we can look up all the Bernoulli numbers we need and write down the polynomial explicitly. For example:
For no obvious reason, let
On the one hand, we have
and so the coefficient of is , more or less the sum we want.
On the other hand, using the definition of the Bernoulli numbers,
so the coefficient of is
Combining these two results, we get
Observe that we need not use the Bernoulli numbers directly; any way we can extract the Taylor coefficients of will give us the same results. In practical terms, we can hand to a computer algebra system and it will print out the needed formula.
This proof is given as Exercise 2.23 in http://www.math.upenn.edu/ wilf/DownldGF.htmlgeneratingfunctionology.
|Title||sum of th powers of the first positive integers|
|Date of creation||2013-03-22 14:14:38|
|Last modified on||2013-03-22 14:14:38|
|Last modified by||mathcam (2727)|