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I am tasked to find the generating function of the sequence $(a_0, a_1, a_2, \dots)$ where $a_n = n2^n$

Here is how I approached it: First, I wrote out the first few terms of the sequence, $(0, 2, 8, 24, 64)$.

Then, using the definition of a generating function, set up this summation:

$$ \sum_{n=0}^{}n2^{n}x^n = \sum_{n=0}^{}n(2x)^{n}$$

However, I am stuck here. I am not sure if this is the right start, but it seems promising.

zzzz
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1 Answers1

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Let $f(x)=\sum_{n=1}^{}n(x)^{n-1}$. Then integrate it.
$\int f(x)dx=\sum_{n=1}^{}(x)^{n}=\frac{x}{1-x}$.
$f(x)=\frac{d}{dx}\frac{x}{1-x}=\frac{1}{(1-x)^2}$
$\frac{2x}{(1-2x)^2}=2xf(2x)= \sum_{n=0}^{}n(2x)^{n}$.

MH.Lee
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