I've got stuck in my homework, and I don't really know if I should look for another solution or I just don't have enough knowledge to do it. So I need to find generating function (as in title) for: $$ b_{n+1} = b_{n} + \frac{1}{n+1}b_{n}, b_0 = 2 $$ And I get to the point: $$ B(x) = xB(x) + \sum_{n=1}^{\infty} (\frac{1}{n}b_{n-1}x^n) + 2 $$ And now I'm stuck. I know the sum above is integral of B(x), but is it possible to get something more of it, maybe by convolution? I would be grateful for some hint, or just tip that it is impossible to get compact form of it.
//Edited typo, sorry.
This is a linear first order recurrence. If you have: $$ x_{n + 1} - a_n x_n = f_n $$ you can divide by $a_n a_{n - 1} \ldots a_0$ and get a new recurrence in $\dfrac{x_n}{a_{n - 1} a_{n - 2} \ldots a_0}$ with a left hand side that telescopes nicely, and a simple sum at the right.