110.406 ANALYSIS II 3 March 1994

Mid-Term Examination

60 minutes. Open book. Notes are allowed.

100 points, 25 per question; partial credit is available.

Use only the officially provided color-coded blue books.

Begin each of the four questions on a fresh page and number it clearly in the margin.

Give reasons and show your working; just answering YES or NO is not enough.

1. (a) Find the power series expansion of log(1 + x) and determine its interval of convergence.

(b) Use (a) to find the power series expansion of the function f defined by

f(x) = ∫_{0}^{x} \frac{log(1 + t)}{t}dt .

What is its interval of convergence? [Warning: Do not attempt to evaluate this integral!]

2. Denote by X the unit interval [0, 1] in R. Define the function f_{n} on X for n = 1, 2, 3, … by f_{n}(x) = x(1 − x)^{n}.

(a) Compute the norm ∥f_{n}∥ of f_{n} in the space C(X) of continuous functions on X.

(b) Does the sequence {f_{n}(x)} converge uniformly on X? If so, to what?

(c) Does the series ∑_{n=1}^{∞} f_{n}(x) converge uniformly on X? If so, to what?

3. Put g_{n}(x) = e^{x}/n^{2} for all real x and positive integers n. The series ∑_{n} g_{n}(x) converges for all x, but not uniformly. Let E be any interval (finite or infinite, open or closed) in the real line. Find the necessary and sufficient condition on E for ∑_{n} g_{n}(x) to converge uniformly for x ∈ E.

4. Given any complex-valued continuous function h(x), defined for all real x, is it possible to approximate h by a function of the form ∑_{n=0}^{N} a_{n} sin^{n} x, where each a_{n} is a complex number:

(a) uniformly for x ∈ [0, π/2]?

(b) uniformly for x ∈ [0, π]?

(c) uniformly for x ∈ [−π/2, π/2]?

THIS QUESTION SHEET MAY BE RETAINED JMB