Math205b Winter Term 2003

Assignment 12

1. Determine the convergence of the following series:

∑_{k=1}^{∞} \frac{e^{i \frac{π}{2} k}}{k} , ∑_{k=1}^{∞}(−1)^{k}(\sqrt{k + 1} − \sqrt{k}) , ∑_{k=1}^{∞} \frac{k!}{k^{k}} .

2. Give an example of (a_{mn})_{m,n∈ℕ} for which

∑_{m=1}^{∞}(∑_{n=1}^{∞} a_{mn}) ≠ ∑_{n=1}^{∞}(∑_{m=1}^{∞} a_{mn}) .

3. Find a sequence of functions (f_{n})_{n∈ℕ} in R([0, 1], ℝ) which converges pointwise to zero but for which

∫_{0}^{1} f_{n}(x) dx ↛ 0 (n → ∞) .

Also find a sequence which does not converge to zero pointwise but for which

∫_{0}^{1} f_{n}(x) dx → 0 (n → ∞) .

4. For α ∈ (0, 1) let

C^{α}([0, 1], 𝕂)

:= \{f ∈ C([0, 1], 𝕂) | [f]_{α} = sup_{x≠y∈[0,1]} \frac{|f(x) − f(y)|}{|x − y|^{α}} < ∞\}

be the space of Hölder continuous real- or complex-valued functions defined on [0, 1]. Show that it is complete w.r.t. ∥ · ∥_{α} = ∥ · ∥_{∞} + [·]_{α}. In other words, prove that any sequence satisfying

∀ε > 0 ∃M ∈ ℕ s.t. ∥f_{n} − f_{m}∥_{α} ≤ ε, m, n ≥ M .

converges to some limit f ∈ C^{α}([0, 1], 𝕂).

5. You ask a question.

The Homework is due Friday, February 7