and for

LHS

RHS 1st term

2nd term

So ; And

I don't know how to turn this into the answer in the book which is

If I've made an error, could someone please point it out?

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- Jul 24th 2010, 03:08 PMoldguynewstudentOrdinary generating function
and for

LHS

RHS 1st term

2nd term

So ; And

I don't know how to turn this into the answer in the book which is

If I've made an error, could someone please point it out? - Jul 24th 2010, 06:38 PMSoroban
Hello, oldguynewstudent!

Quote:

I cranked out the first few terms and looked for a pattern.

. .

Therefore: .

- Jul 24th 2010, 09:46 PMroninpro
Your answer looks fine. The last thing you have to do is find the Taylor series for .

You can start with the geometric series

Then,

Now, differentiating both sides,

Finally, dividing both sides by ,

We see that for (You can shift the index by one to get book's solution.) - Jul 25th 2010, 07:39 AMRenji Rodrigo
a general solution for recurrences in the form

can be found using this trick

write

then use it on the equation

so

is a telescoping summation, finding h(n) we find f(n)

by example, puting a_k=3, we have h(0)=3 , by the initial condition so

h(n)=3(n+1) so . - Jul 25th 2010, 01:54 PMoldguynewstudent
Yes, I see your point, but in the context of ordinary generating functions, I believe this approach would not go over with my professor.