I've been trying to find a way to prove the formula for the bernoulli numbers. I've been trying to prove it by induction because that's pretty much the only proof I've learned so far, the only problem is the m-1 on top in the sigma not'n. I'm not sure what to do with it, would anyone be able to give me a bit of a head start. The simplest way possible would be nice.

Thanks

2. Originally Posted by sugar_babee
I've been trying to find a way to prove the formula for the bernoulli numbers. I've been trying to prove it by induction because that's pretty much the only proof I've learned so far, the only problem is the m-1 on top in the sigma not'n. I'm not sure what to do with it, would anyone be able to give me a bit of a head start. The simplest way possible would be nice.

Thanks
see if these help:

Faulhaber's formula

Bernoulli number

Geometric progression

3. I've looked at those, what I dont get from that is how they got from the first equation to the one below. And I just really want to know what to do with the m-1 on the left side of the equation. I expanded it to be 0^n + 1^n+...+(m+1)^n What I need to figure out is a way to use mathematical induction which means i need to make a set from the bernoulli equation and then claim 1 is in the set, prove it, then k+1 is in the set. How/where would I sub in the 1 for that? Basically all I want to do is a left-side, right-side proof

4. Originally Posted by sugar_babee
I've been trying to find a way to prove the formula for the bernoulli numbers. I've been trying to prove it by induction because that's pretty much the only proof I've learned so far, the only problem is the m-1 on top in the sigma not'n. I'm not sure what to do with it, would anyone be able to give me a bit of a head start. The simplest way possible would be nice.

Thanks
This is the (well one at least) definition of the Bernoulli numbers - what are
you trying to proove.

Do you have another definition of the Bernoulli numbers that you wish to
prove this formula from?

RonL