Maclaurin Series Question

[PeacefulChaos]

I have no idea where to even begin with this problem. Can someone explain to me where they came up with the equation that the derivatives satisfy? I tried to work out the derivatives on my own to see if they do in fact satisfy the said equation but I get lost where they got the coefficients of the derivatives.

Any help is appreciated, thanks. Oh, and this is from an HL exam but I do not know what year...

[sweetnsimple786]

While I've done MacLaurin series, I haven't seen something like this, so here's my shot at an explanation. I think I might be on the right track, but I don't have it all the way.

so if a_n is the coeffiecient, that means that a_n = f/n!

and a_n+2 = f⁽ⁿ⁺²⁾/(n+2)! = f⁽ⁿ⁺²⁾/n!(n+1)(n+2) [see where the (n+1)(n+2) might come in now?]

So what I said was that

a_n = f/n!

a_n+2 = a_n / (n+1)(n+2) --> (n+1)(n+2)a_{n+2 = an}

_{Now where does that n² term come in? This is what I'm not sure about. But I think it has to do with the expression for the derivatives. It seems that if you start out at the 5th derivative, the 4th derivative will have one less power of n. The 3rd derivative will have 2 less powers of n in comparison with the 5th derivative. [shown by the n⁰ term in the f⁽ⁿ⁺²⁾ and the n¹ term in the f⁽ⁿ⁺¹⁾ and the n² term in the f]}

_{Maybe you can see the step I'm missing. Or maybe I'm completely wrong... Ah well. G'luck.}

[PeacefulChaos]

Ahh, I see where some of it came from now. I can get myself to where you encountered the problem.

I have a lot of other review packets to do so I'm going to move on and come back later.

Thanks for the help and I'll post if I find the solution.