- Thread starter
- #1

#### OhMyMarkov

##### Member

- Mar 5, 2012

- 83

Suppose $f(x)$ can be written as $f(x)=P_n(x)+R_n(x)$ where the first term on the RHS is the Taylor polynomial and the second term is the remainder.

If the sum $\sum _{n=0} ^{\infty} = c_n x^n$ converges for $|x|<R$, does this mean I can freely write $f(x)=\sum _{n=0} ^{\infty} = c_n x^n$?

Can I also use the fact that the sum of continuous functions over a domain (in this case, $|x|<R$) is continuous, and that the sum of differentiable functions over a domain is differentiable?