# 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|.

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...

• Further to the answer posted by @nala, if power series converges absolutely and so does the product of two power series. Therefore, rearrangement of terms doesn't change the sum of the series.
– Koro
May 11 '20 at 12:10
• An other question, now how can I find the first 4 addends of the Taylor serie of f(x)=(e^x)/(3+2x) in the point x_0=0? May 11 '20 at 13:43
• Write series for $e^x$, which is $1+x +x^2/2!+ x^3/3!+...$ and then for $\frac{1}{1+(2x/3)}$. Multiply them and collect the terms of x with power upto 3 to get first four terms.
– Koro
May 11 '20 at 15:22

$$\left(\displaystyle\sum_{n=0}^{+\infty} a_n x^n\right) \cdot \left(\displaystyle\sum_{n=0}^{+\infty} b_n x^n\right) = \displaystyle\sum_{n=0}^{+\infty} c_n x_n$$, where each of the $$c_n$$ is defined as $$c_n = \displaystyle\sum_{j = 0}^{n} a_j b_{n-j}$$.
• Are you supposed to use a Cauchy product? :) Cause if so, try and see $\frac{1}{3+2x}$ as $\frac{1}{3}\cdot\frac{1}{1 + 2x/3}$, and then expend it as a geometric series :)