/sci/ - Science & Math

[Anonymous No.6924653]

Was wondering if any of you could help me out some some simple recursion that I can't seem to wrap my head around

And not to beat around the bush, yes this is a homework problem, but I'm not after an answer (if that changes anything) just need a clarification

the recurrence is a(n) = na(n-1) + n!

I've thought about using generating functions but the summation of (n!)(x^n) does not converge

and so that leaves me with finding a particular solution using factorials, but I've only ever found a particular solution using exponential and polynomials, what the hell do you do with a factorial?

[Anonymous No.6924658]

with some*

[Anonymous No.6924673]

a(n) = na(n-1) + n!
a(n-1) = (n-1) a(n-2) + (n-1)!
n a(n-1) = n (n-1) a(n-2) + n!

Subract this equation from the first one:
a(n) = 2 n a(n-1) - n (n-1) a(n-2)

You should be able to solve this.

[Anonymous No.6924676]

>>6924653

try using rising factorials

[Anonymous No.6924684]

>>6924673

f(x) = f(x-1)+x
f*x = f*x-f+x
0 = 1/fx - 1/x + x
x/x + x*x = 1/f
1+x^2 = 1/f
f=1/(1+x^2)

is this your idea of how math works

[Anonymous No.6924691]

>>6924684
Are you stupid?

[Anonymous No.6924695]

the answers
a(n) = n!*a(0) + (n+1)!

[Anonymous No.6924701]

The fastest way to solve this by hand is to divide by n! and then solve the recursion for b(n) := a(n)/n!. But you can also do it like this >>6924673

[Anonymous No.6924703]

>>6924673
damn, that's a smart idea thanks

>>6924676
I thought about that after searching wolframalpha, but we've never even talked about that in class so my professional might be a little skeptical