Hello, oldguynewstudent!
I cranked out the first few terms and looked for a pattern.
. .
Therefore: .
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 .