Basic Analysis I — Real Analysis · Problem 6.1.14

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · geometric-series-uniform-convergence

Problem

(requires \S2.6): In Example 6.1.3 we saw ∑k=0∞xk\sum_{k=0}^\infty x^k converges pointwise to 11−x\frac{1}{1-x} on (−1,1)(-1, 1). (a)) Show that whenever 0≤c<10 \leq c < 1, the series ∑k=0∞xk\sum_{k=0}^\infty x^k converges uniformly on [−c,c][-c, c]. (b)) Show that the series ∑k=0∞xk\sum_{k=0}^\infty x^k does not converge uniformly on (−1,1)(-1, 1).

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