Question

Exercise 1 (a) Prove that the series \sum_{k=1}^{\infty} \frac{1}{(1+\sqrt{x})^k} is, for each \delta > 0, uniformly convergent on D_\delta = [\delta, \infty). (b) Show that \sum_{k=1}^{\infty} \frac{1}{(1+\sqrt{x})^k} is not uniformly convergent on (0, \infty). (c) Compute the limit \sum_{k=1}^{\infty} \frac{1}{(1+\sqrt{x})^k}, show that this limit is uniformly continuous on $[\delta, 1]$ for $\delta \in (0, 1)$, but not uniformly continuous on $(0, 1]$.

          Exercise 1 (a) Prove that the series \sum_{k=1}^{\infty} \frac{1}{(1+\sqrt{x})^k} is, for each \delta > 0, uniformly convergent on D_\delta = [\delta, \infty).
(b) Show that \sum_{k=1}^{\infty} \frac{1}{(1+\sqrt{x})^k} is not uniformly convergent on (0, \infty).
(c) Compute the limit \sum_{k=1}^{\infty} \frac{1}{(1+\sqrt{x})^k}, show that this limit is uniformly continuous on $[\delta, 1]$ for $\delta \in (0, 1)$, but not uniformly continuous on $(0, 1]$.