Suppose that $$\sum\limits_{n=0}^\infty a_nx^n \ \ and \ \ \sum\limits_{n=0}^\infty b_nx^n$$ $R$ and $S$ respectively.Let $U$ be the radius of convergence of $$\sum\limits_{n=0}^\infty c_nx^n$$ where for each $n$ $$c_n=\sum\limits_{k+l=n}^{} a_kb_l$$ Show that $U \geq min\{R,S\}$.
Hint If $|x|<min\{R,S\}$ then the sequence of partial sums $$A_N := \sum\limits_{n=0}^N |a_nx^n| \ \ and \ \ B_N := \sum\limits_{n=0}^N |b_nx^n|$$ both converge. Hence so does their product. Deduce that $\sum\limits_{n=0}^\infty |c_nx^n|$ converges absolutely.

My Attempt

Now just for me to get to grips with the definition of $c_n$ I tried a sample case $n=3$ $$\sum\limits_{n=0}^3 a_nx^n=a_0+a_1x+a_2x^2+a_3x^3$$ $$\sum\limits_{n=0}^3 b_nx^n=b_0+b_1x+b_2x^2+b_3x^3$$
Now for n=3 $c_n$ would be $$\sum\limits_{k+l=3}^{} a_kb_l=a_0b_0+a_1b_0+a_0b_1+a_0b_2+a_1b_1+a_2b_0+a_0b_3+a_1b_2+a_2b_1+a_3b_0$$
and so $$\sum\limits_{n=0}^3 c_nx^n=(a_0b_0)+(a_1b_0+a_0b_1)x+(a_0b_2+a_1b_1+a_2b_0)x^2+(a_0b_3+a_1b_2+a_2b_1+a_3b_0)x^3$$
Now to use the hint; if $|x|<min\{R,S\}$ then the sequence of partial sums converge i.e; $$A_N := \sum\limits_{n=0}^N |a_nx^n|= a_0+a_1x+a_2x^2+a_3x^3+...+a_nx^n \ \ converges$$ $$B_N := \sum\limits_{n=0}^N |b_nx^n|=b_0+b_1x+b_2x^2+b_3x^3+...+b_nx^n \ \ converges$$
and so does their product;$$A_N.B_N := \sum\limits_{n=0}^N |a_nx^n|.\sum\limits_{n=0}^N |b_nx^n|=a_0b_0+a_0b_1x+a_0b_2x^2+a_0b_3x^3+a_1b_0x+a_1b_1x^2+a_1b_2x^3+a_1b_3x^4+a_2b_0x^2+a_2b_1x^3+a_2b_2x^4+a_2b_3x^5+a_3b_0x^3+a_3b_1x^4+a_3b_2x^5+a_3b_3x^6+.......$$

From this we can see, for any $n \in \mathbb{N}$ $$ \sum\limits_{n=0}^n |c_nx^n| \leq A_N.B_N := \sum\limits_{n=0}^N |a_nx^n|.\sum\limits_{n=0}^N |b_nx^n|$$ and so by comparison $$ \sum\limits_{n=0}^n |c_nx^n|$$ converges absolutely which implies that $$\sum\limits_{n=0}^\infty c_nx^n$$ converges for $|x|<min\{R,S\}$ and so the radius of convergence $U \geq min\{R,S\}$

Is this proof correct?? Any help would be much appreciated.


By the theorem of Cauchy-Hadamard, for any $ε>0$ one has $$ |a_n|\le(1+ε)^nR^{-n}\text{ and }|b_n|\le(1+ε)^nS^{-n} $$ so that, again for almost all $n\in\Bbb N$ \begin{align} |c_n|&\le (1+ε)^n(R^{-n}+R^{-n+1}S^{-1}+R^{-n+2}S^{-2}+...+R^{-1}S^{-n+1}+S^{-n}) \\[0.3em] &\le n(1+ε)^n\max(R^{-1},S^{-1})^n=\frac{n(1+ε)^n}{\min(R,S)^n} \end{align} Consequently, $$ \limsup_{n\to\infty}\sqrt[n]{|c_n|}\le\frac{(1+ε)}{\min(R,S)} $$ with the corresponding consequences for the radius of convergence.


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