<s>[INST] Textbook Proposition on Product of Real Analytic Functions 
Let
  \begin{align*}
  \sum\limits_{j=0}^\infty a_j (x-c)^j && \sum\limits_{j=0}^\infty b_j (x-c)^j \\
\end{align*}
be two power series with intervals of convergence $\mathcal{C}_1$ and $\mathcal{C}_2$ centered on at $c$. Let $f_1(x)$ be the function defined by the first series on $\mathcal{C}_1$ and $f_2(x)$ the function defined by the second series on $\mathcal{C}_2$. Then, on their common domain $\mathcal{C} = \mathcal{C}_1 \cap \mathcal{C}_2$, it holds that
  \begin{align*}
  f(x) \cdot g(x) &= \sum\limits_{m=0}^\infty \sum\limits_{j+k=m} (a_j \cdot b_k) (x-c)^m \\
\end{align*}
Proof: Let
  \begin{align*}
  A_N &= \sum\limits_{j=0}^N a_j (x-c)^j & B_N &= \sum\limits_{j=0}^N b_j (x-c)^j \\
\end{align*}
be respectively, the Nth partial sums of the power series that define $f$ and $g$. Let
  \begin{align*}
  D_N &= \sum\limits_{m=0}^N \sum\limits_{j+k=m} (a_j \cdot b_k) (x-c)^m &  
  R_N &= \sum\limits_{j=N+1}^\infty b_j (x-c)^j \\
\end{align*}
We have:
  \begin{align*}
  D_N &= a_0 B_N + a_1 (x-c) B_{N-1} + \cdots + a_N (x-c)^N B_0 \\
  &= a_0 (g(x) - R_N) + a_1 (x-c) (g(x) - R_{N-1}) + \cdots + a_N (x-c)^N (g(x) - R_0) \\
\end{align*}
  [snip]

I've transcribed the above from my Real Analysis textbook and stopped at the part I'm having trouble understanding. How do you get from the first definition of $D_N$ to the second one? They seem like different equations. I've tried to do sequence manipulations to get from one definition to the other, but I've not had success.
 [/INST] You can actually prove this by induction

Works for $N = 1$

\begin{eqnarray}
D_1 &=& a_0b_0 + (a_0b_1 + a_1 b_0)(x - c) \\ &=& a_0 [b_0 + b_1(x - c)] + a_1[b_0(x - c)] \\
&=& a_0 B_1 + a_1 B_0
\end{eqnarray}

Assume that it works for $N - 1$

$$
\sum_{m = 0}^{N - 1}\sum_{j + k = m}a_j b_k(x - c)^m = a_0 B_{N - 1}
+ a_1(x - c) B_{N - 2}  + \cdots + a_{N-1}(x - c)^{N-1}B_0 \tag{1}
$$

Now let's prove it for $N$

\begin{eqnarray}
\sum_{m = 0}^{N}\sum_{j + k = m}a_j b_k(x - c)^m &=& \sum_{m = 0}^{N - 1} \sum_{j + k = m}a_j b_k(x - c)^m + \sum_{j + k = N}a_j b_k(x - c)^N \\
&\stackrel{(1)}{=}& a_0 B_{N - 1}
+ a_1(x - c) B_{N - 2}  + \cdots + a_{N-1}(x - c)^{N-1}B_0 \\
&& + [a_0 b_N + a_1 b_{N - 1} + \cdots + a_N b_0](x - c)^N \\
&=& a_0 \left[ B_{N - 1} + b_N(x - c)^N \right] + a_1(x - c) \left[ B_{N - 2} + b_{N - 1}(x - c)^{N -  1} \right] \\
&& + \cdots + a_{N}(x - c)^{N}[b_0] \\
&=&a_0 B_N + a_1(x - c) B_{N - 1} + \cdots + a_{N}(x - c)^{N}B_0 \tag{2}
\end{eqnarray}
</s>