Question 9.T.15: Let R be the radius of convergence of the power series ∑anx^...

For any x ∈ [−r, r],

$\left|a_{n}x^{n}\right| ≤ |a_{n}| r^{n},  n ∈ \mathbb{N}_{0}.$

Since r ∈ (−R, R), the (numerical) series $\sum{\left|a_{n}\right|r^{n}}$ is convergent. By the M-test, with $M_{n} = |a_{n}|r^{n},$ the series $\sum{a_{n}x^{n}}$ is uniformly convergent on [−r, r].