The multiplication of two power series
Let R be a real positive number,$$f(x)=\sum_{n=0}^\infty a_n x^n \\ g(x)=\sum_{n=0}^\infty b_n x^n $$where $|x|<R$.
I have to give the first 4 addends depending on $\{a_n\}$ and $\{b_n\}$.
I know that$$f(x)= a_0 + a_1 x + a_2 x^2 + ... \\ g(x)= b_0 + b_1 x + b_2 x^2 + ... $$I've multiplied some elements and I think that I've got the result:$$f(x)g(x)= a_0b_0+(a_0b_1 + a_1b_0)x+(a_0b_2 + a_1b_1 + a_2b_0)x^2+(a_0b_3 + a_1b_2 + a_2b_1 + a_3b_0)x^3+...$$But I think that's not the good way to do it, I suppose there is an other better way to do it. Anyway, I don't know if my answer is ok...
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$\begingroup$this kind of product is called the Cauchy product of two series :) You got the hang of it for the first few terms, now, by induction, it generalizes to:
$\left(\displaystyle\sum_{n=0}^{+\infty} a_n x^n\right) \cdot \left(\displaystyle\sum_{n=0}^{+\infty} b_n x^n\right) = \displaystyle\sum_{n=0}^{+\infty} c_n x_n$, where each of the $c_n$ is defined as $c_n = \displaystyle\sum_{j = 0}^{n} a_j b_{n-j}$.
This coincides with the first few terms you computed :). You can see more on the wikipedia page on Cauchy product.
PS: try to LaTeX your question next time, it makes it clearer to read :)
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