TITLE: Generating functions for $n*2^n$ & the seq a0+a1+a2+...
QUESTION [1 upvotes]: 1) What is the generating function of $a_n = n2^n, n\geq0$?
My answer: $f(x) = \sum a_nx^n = \sum n2^nx^n = \sum n(2x)^n$, but I have no idea where to go from here.
2) Let the sequence $s_n = a_0 + a_1 + a_2 + ... + a_n$. What is its generating function?
My answer: $f(x) = \sum a_nx^0 = \sum a_n ....$ this is where I am stuck.

REPLY [1 votes]: Hint: We want $\sum n(2x)^n$. Note that 
$$1+2x+(2x)^2+ (2x)^3+\cdots=\frac{1}{1-2x}$$
for suitable $x$. Differentiate term by term, and you are almost there.
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