Power series may be differentiated term by term inside their radius

Fix $c$ with $\mathrm{\mid }c\mathrm{\mid }<R_0$ and choose $R$ with $\mathrm{\mid }c\mathrm{\mid }<R<R_0$. For $0<\mathrm{\mid }h\mathrm{\mid }<R-\mathrm{\mid }c\mathrm{\mid }$, the identity $(c+h{)}^n-c^n=h{\sum }_{j=0}^{n-1}(c+h{)}^{n-1-j}{c}^j$ for $n\ge 1$ and [L1] yield $\mathrm{\mid }((c+h{)}^n-c^n)\mathrm{/}h\mathrm{\mid }\le n{R}^{n-1}$. Thus the tail of the difference-quotient series beyond degree $N$ is at most ${\sum }_{n>N}n\mathrm{\mid }{a}_n\mathrm{\mid }{R}^{n-1}$, uniformly for these $h$. The tail of the proposed derivative series at $c$ has the same bound.

For fixed $N$, the difference quotient of the polynomial ${\sum }_{n=0}^N{a}_n{x}^n$ tends to ${\sum }_{n=1}^{N}n{a}_n{c}^{n-1}$ as $h\to 0$ by the binomial identity. Given $\epsilon >0$, choose $N$ so both tails bounded above are below $\epsilon \mathrm{/}3$, then choose $h$ small enough that the finite polynomial quotient differs from its derivative by less than $\epsilon \mathrm{/}3$. The full difference quotient therefore differs from ${\sum }_{n\ge 1}n{a}_n{c}^{n-1}$ by less than $\epsilon$, proving the derivative formula.