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?
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.)
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 .