Understanding Faulhaber's Formula for Sum of Powers of n Integers

[plzhelpme]

hi everybody, I have a question in math to figure out the general term for the sum of the pth powers of n integers. I found a formula called faulhabers formula to do this question, but I do not understand the method behind it. can someone please help me?

 

[shmoe]

You can prove it using Euler-MacLaurin summation, where you essentially rewrite the sum as a Stieltjes integral then integrate by parts enough time that the remaining integrand is zero (the p+1 derivitative of x^p vanishes). Look up Euler-MacLaurin summation (lot's of references).

Or are you trying to understand how to apply Faulhaber's in a given situation? It's just plug and chug, but you'll need to know what the Bernoulli numbers are (again, lots of references)

 

[tacman]

Sure, I would be happy to help you understand Faulhaber's formula for finding the sum of powers of n integers. Faulhaber's formula is a mathematical formula that helps us find the sum of the pth powers of n integers, where p is any positive integer and n is the number of terms in the series.

The formula is given by:
∑k^p = 1^p + 2^p + 3^p + ... + n^p = (n(n+1)/2)^p

To understand this formula, we first need to understand the concept of summation. Summation is a mathematical operation that adds together a sequence of numbers. In this case, we are adding together the pth powers of n integers.

Now, let's break down the formula. The symbol ∑ represents summation, and the letter k is the index of summation, which means it takes on different values as we add up the terms. In this case, k takes on the values of 1, 2, 3, ..., n.

The term (n(n+1)/2) is known as the triangular number. It is derived from the fact that the sum of the first n natural numbers is equal to (n(n+1))/2. For example, the sum of the first 5 natural numbers (1+2+3+4+5) is equal to (5(5+1))/2 = 15.

Finally, the term (n(n+1)/2)^p represents the sum of the pth powers of the first n natural numbers. This is because we are essentially multiplying the triangular number by itself p times.

So, to find the sum of the pth powers of n integers, we simply plug in the value of n into the formula and raise it to the power of p. For example, if we want to find the sum of the 3rd powers of the first 5 natural numbers, we would plug in n=5 and p=3 into the formula and get (5(5+1)/2)^3 = 225.

I hope this explanation helps you understand Faulhaber's formula better. Remember, practice makes perfect, so try out some examples to solidify your understanding. Good luck!