Example 8.3.4 (a): Geometric Series Function
f(x) = fn(x) = xnis continuous on [-r, r]. Can you find a simpler expression for f?
If -1 < -r x r < then || xn ||[-r, r] = rn. Since rn < the Weierstrass convergence theorem applies immediately to show that the series represents a continuous function.
Of course we have seen this series before and know it as geometric series with limiting function 1/1-x. But this time we know as an application of Weierstrass' theorem that the series is a continuous function, whether it has a simpler representation or not.