Suppose you have a complex power series with radius of convergence which converges at some point on the boundary of the disk . By shifting and rotating the axes, you obtain another power series with radius of convergence which converges at the point . (If then .)
Now you need Abel's test: Let be a sequence of complex functions on a set and let be a decreasing sequence of non-negative functions on . If the series converges uniformly on and if there is a constant such that for every and for all non-negative integers , then converges uniformly on .
In this case, let , and for let and .
By hypothesis, the series is convergent, so converges uniformly on . Also for all and for all .
By Abel's test, the series converges uniformly on .
Thus the original series is uniformly convergent on the radius of from to .
Will this do?