## Example 8.3.4 (a): Geometric Series Function

Define

*f*for_{n}(x) = x^{n}*x [-r, r]*, where*0 < r < 1*. Then the functionis continuous onf(x) = f_{n}(x) = x^{n}

*[-r, r]*. Can you find a simpler expression for*f*?If
*-1 < -r x r < * then
*|| x ^{n} ||_{[-r, r]} = r^{n}*. Since

*r*the Weierstrass convergence theorem applies immediately to show that the series represents a continuous function.

^{n}<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.