Exercises
[1KQ]Prerequisites:[1K9].Consider power series
\[ f(x)=∑_{n=0}^∞ a_ n x^ n ~ ~ ,~ ~ g(x)=∑_{m=0}^∞ b_ m x^ m~ , \]with non-zero radius of convergence, respectively \(r_ f\) and \(r_ g\).
Show that the product function \(h(x)=f(x)g(x)\) can be expressed in power series
\[ h(x)=∑_{k=0}^∞ c_ k x^ k \]where
\[ c_ k = ∑_{j=0}^ k a_ j b_{k-j}~ ; \]with radius of convergence \(r_ h≥\min \{ r_ f, r_ g\} \). (Note the similarity with Cauchy’s product, discussed in section [0CN])
Can it happen that \(r_ h{\gt}\min \{ r_ f, r_ g\} \)?
Solution 1