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Example 8.3.4 (a): Geometric Series Function

Define fn(x) = xn for x [-r, r], where 0 < r < 1. Then the function
f(x) = fn(x) = xn
is 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.

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