Sample Midterm Questions Math 127B. Winter, 2005

Closed Book. No calculators. Give complete proofs of all your answers.

1. Consider the sequence (f_{n}) of functions f_{n} : ℝ → ℝ defined by

f_{n}(x) = \frac{nx}{\sqrt{1 + n^{2}x^{2}}} .

Find the pointwise limit of this sequence as n → ∞. Does the sequence converge uniformly on ℝ? Justify your answer.

2. Let

f_{n}(x) = \frac{nx + sin(nx^{2})}{n}.

Prove that the following limit exists, and compute its value:

lim_{n→∞} ∫_{0}^{1} f_{n}(x) dx.

3. Prove that the following series

f(x) = ∑_{n=1}^{∞} \frac{n^{2} + x^{4}}{n^{4} + x^{2}}

converges to a continuous function f : ℝ → ℝ.

4. Determine the radius of convergence R of the power series

f(x) = ∑_{n=0}^{∞} \frac{(−1)^{n}}{2n + 1}x^{2n+1}

= x − \frac{x^{3}}{3} + \frac{x^{5}}{5} − \frac{x^{7}}{7} + ….

Where does the series converge? Prove that

f′(x) = \frac{1}{1 + x^{2}} in |x| < R.

5. Suppose that (f_{n}) is a sequence of functions f_{n} : [−1, 1] → ℝ that converges uniformly on [−1, 1] to a function f : ℝ → ℝ. If the limit

lim_{x→0} f_{n}(x) = a_{n}

exists for each n ∈ ℕ, and the limit

lim_{n→∞} a_{n} = a

exists, prove that lim_{x→0} f(x) exists, and

lim_{x→0} f(x) = a

Give a counter-example to show that this result need not be true if (f_{n}) converges to f pointwise, but not uniformly.