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What are the Additional And Multiplication Rules of Formal Power Series?
in A Level Maths by Diamond (49,543 points) | 12 views

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Let $A(x)$ and $B(x)$ be formal power series. \\
(Addition)
\begin{gather*}
A(x) + B(x) = (a_0 + b_0) + (a_1 + b_1)x + (a_2 + b_2)x^2 + ... = \sum_{k = 0}^{\infty}(a_k + b_k)x^k
\end{gather*}
(Multiplication)
\begin{align*}
A(x)B(x) &= \Bigg(\sum_{i \geq 0}a_ix^i\Bigg)\Bigg(\sum_{j \geq 0}b_jx^j\Bigg) \\
&= \sum_{i \geq 0}\sum_{j \geq 0}a_ib_jx^{i+j} \\
&= \sum_{n \geq 0}\sum_{k = 0}^{n}a_kb_{n-k}x^n
\end{align*}

Theorem:
Let $A(x) = a_0 + a_1x + a_2x^2 + ..., P(x) = p_0 + p_1x + p_2x^2 + ...$, and $Q(x) = 1 - q_1x - q_2x^2 - ...$ be formal power series. Then
\begin{gather*}
Q(x)A(x) = P(x) \\
\iff \\
a_n = p_n + q_1a_{n-1} + q_2a_{n-2} + .... + q_na_0
\end{gather*}

Let $P(x)$ and $Q(x)$ be formal power series.
\begin{gather*}
Q(0) \neq 0 \implies \exists!A(x) \quad Q(x)A(x) = P(x)
\end{gather*}
by Diamond (49,543 points)

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