The series gives the sum of the powers of the first positive numbers, where and are positive integers. Each of these series can be calculated through a closed-form formula. The case is famously said to have been solved by Gauss as a young schoolboy: given the tedious task of adding the first positive integers, Gauss quickly used a formula to calculate the sum of
The formulas for the first few values of are as follows:
Faulhaber's formula, which is derived below, provides a generalized formula to compute these sums for any value of
Let The elementary trick for solving this equation (which Gauss is supposed to have used as a child) is a rearrangement of the sum as follows:
Grouping and adding the above two sums gives
Find the sum of the first positive integers.
Plugging in our equation,
which implies our final answer is 5050.
Show that the sum of the first positive odd integers is
There are several ways to solve this problem. One way is to view the sum as the sum of the first integers minus the sum of the first even integers. The sum of the first even integers is times the sum of the first integers, so putting this all together gives
Even more succinctly, the sum can be written as
In a similar vein to the previous exercise, here is another way of deriving the formula for the sum of the first positive integers. Start with the binomial expansion of
Rearrange the terms as below:
Now sum both sides:
The left sum telescopes: it equals The right side equals which gives so
This technique generalizes to a computation of any particular power sum one might wish to compute.
Continuing the idea from the previous section, start with the binomial expansion of
Rearrange the terms:
As before, summing the left side from to yields This gives
Find the sum of the squares of the first positive integers.
As in the previous section, let Then the relevant identity, derived in the same way from the binomial expansion, is
This recursive identity gives a formula for in terms of for It is the basis of many inductive arguments. In particular, the first pattern that one notices after deriving for is the leading terms Here is an easy argument that the pattern continues:
For a positive integer is a polynomial of degree in Its leading term is
Induction. The statement is true for and now suppose it is true for all positive integers less than Then solve the above recurrence for to get
where the are some rational numbers.
Now by the inductive hypothesis, all of the terms except for the first term are polynomials of degree in so the statement follows.
Note the analogy to the continuous version of the sum: the integral The lower-degree terms can be viewed as error terms in the approximation of the area under the curve by the rectangles of width and height
Having established that the obvious question is whether there is an explicit expression for the lower terms. It turns out that the terms can be expressed quite concisely in terms of the Bernoulli numbers, as follows:
That is, if is a positive integer, the coefficient of in the polynomial expression for the sum is
This can be read off directly from Faulhaber's formula: the term is and the term is
and since this simplifies to
To compute using Faulhaber's formula, write
and use to get
This happens to factor as
Note that the sign only affects the term when because the odd Bernoulli numbers are zero except for
The proof of the theorem is straightforward (and is omitted here); it can be done inductively via standard recurrences involving the Bernoulli numbers, or more elegantly via the generating function for the Bernoulli numbers.