# Function represented by power series

## Homework Statement

If a function f is represented by the power series ∑(k=0 to ∞) a_k(x-a)^k, with a radius of convergence )<R<∞, then f is continuous on the interval (a-R, a+R)

## The Attempt at a Solution

I don't know if my proof is loose at some point or not, because it seems so easy. Is there anything that is obviously not right about my proof? Thanks very much

So choose x=m in (a-R, a+R), the series converges on the interval (a-R,a+R)
then we would have
lim(x->m) ∑(k=0 to ∞) a_k(x-a)^k = ∑(k=0 to ∞) (a_k)lim(x->m) (x-a)^k = ∑(k=0 to ∞) (a_k)(m-a)^k = f(m).

So f is continuous on any arbitrary m on (a-R, a+R) therefore ∑(k=0 to ∞) a_k(x-a)^k is continuous on (a-R, a+R)

Related Calculus and Beyond Homework Help News on Phys.org
Stephen Tashi
$$\lim_{x \to m} \lim_{n \to \infty} \sum_{k=0}^n a_k (x-a)^k = \lim_{n \to \infty} \lim_{x \to m} \sum_{k=0}^n a_k (x-a)^k$$.
Consider $x_{m,n}$ = 0 when n < m and 1 when n >= m. Then $\lim_{n \to \infty} \lim_{m \to \infty} x_{m,n} \neq \lim_{m \to \infty} \lim_{n \to \infty} x_{m,n}$.