I was messing around with some math formulas today and came up with a result that I found pretty neat, and I would appreciate it if anyone could explain it to me.
The formula for an infinite arithmetic sum is n∑i=1ai=n(a1+an)2,
so if you want to find the sum of the natural numbers from 1 to n, this equation becomes n2+n2,
and the roots of this quadratic are at n=−1 and 0. What I find really interesting is that ∫0−1n2+n2dn=−112
There are a lot of people who claim that the sum of all natural numbers is −112, so I was wondering if this result is a complete coincidence or if there's something else to glean from it.
Answer
We have Faulhaber's formula:
n∑k=1kp=1p+1p∑j=0(−1)j(p+1j)Bjnp+1−j, where B1=−12
⟹fp(x)=1p+1p∑j=0(−1)j(p+1j)Bjxp+1−j
We integrate the RHS from −1 to 0 to get
Ip=∫0−1fp(x) dx=(−1)pp+1p∑j=0(p+1j)Bjp+2−j
Using the recursive definition of the Bernoulli numbers,
Ip=(−1)pBp+1p+1=−Bp+1p+1
Using the well known relation Bp=−pζ(1−p), we get
Ip=ζ(−p)
So no coincidence here!
No comments:
Post a Comment