Math205b Winter Term 2003

Midterm Examination

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You have 50 minutes to solve the problems. Good luck!

1. Let (a_{n})_{n∈ℕ} ∈ ℝ^{ℕ} with a_{n} ≥ 0 n ∈ ℕ. Show that

∑_{n=0}^{∞} a_{n} < ∞ ⟺ ∑_{n=0}^{∞} \frac{a_{n}}{1 + a_{n}} < ∞ .

2. Determine convergence or radius of convergence for the following series

∑_{n=2}^{∞} \frac{1}{k^{2} − k} , ∑_{n=0}^{∞} \frac{k^{2}}{2^{k}} , ∑_{n=1}^{∞}(log n) x^{n} .

Then compute the value of the first one.

3. Show that the improper integral

∫_{0}^{∞} \frac{cos(x)}{\sqrt{x}} dx

exists.

4. Assume that f ∈ C^{1}([0, π]) and define

g_{n}(x) = ∫_{0}^{x} sin(n(x − y))f(y) dy , x ∈ [0, π] .

Show that g_{n} converges pointwise to zero. Is the convergence uniform?

5. Let f_{n} ∈ C([0, 1], ℝ) and assume that (f_{n})_{n∈ℕ} converges pointwise and monotonically to f ≡ 0. Prove that the convergence is then already uniform.