Math 312, Intro. to Real Analysis: Midterm Exam #2

Stephen G. Simpson

Friday, March 27, 2009

1. True or False (2 points each)

(a) Every monotone sequence of real numbers is convergent.

(b) Every sequence of real numbers has a lim sup and a lim inf.

(c) Every sequence of real numbers has a monotone subsequence.

(d) Every sequence of real numbers has a convergent subsequence.

(e) If lim inf a_{n} = lim sup a_{n} = α then lim a_{n} = α.

(f) We can find a sequence of real numbers, (a_{n}), such that the subsequential limits of (a_{n}) are exactly the real numbers in the closed interval [−1, 1].

(g) The series 1 − \frac{1}{4} + \frac{1}{9} − \frac{1}{16} + ⋯ is absolutely convergent.

(h) If ∑ |a_{n}| is convergent, then so is ∑ |a_{n}|^{2}.

(i) If ∑ a_{n} is convergent, then so is ∑ a_{n}^{2}.

(j) ∑_{n=1}^{∞} \frac{1}{n^{p}} is convergent for all p ≥ 1.

(k) ∑_{n=1}^{∞} \frac{1}{10^{n}} = \frac{1}{9}.

(l) The function \sqrt{x} is uniformly continuous on [0, ∞).

(m) If a function is uniformly continuous on the interval (a, b] and on the

interval [b, c), then it is uniformly continuous on the interval (a, c).

2. (6 points each) Which of the following series are convergent and/or absolutely convergent? Please indicate which tests you are using and show your work.

(a) 1 − \frac{1}{2} + \frac{1}{3} − \frac{1}{4} + ⋯

(b) ∑_{n=1}^{∞}(\sqrt{n^{2} + n} − n)

(c) ∑_{n=1}^{∞}(\sqrt{n^{2} + 1} − n)

(d) ∑ \frac{1}{1.01^{n} − 1000}
(e) ∑ \frac{1}{n^{2} + \sqrt{n^{2} + 1}}
(f) ∑_{n=1}^{∞} log \frac{n + 1}{n}

3. (8 points) Use algebra plus limit laws to calculate

lim \frac{log \sqrt{e^{(2n+11)/n}}}{sin((4nπ + 5)/16n)}.

In performing this calculation, you may take it for granted that functions such as sin x, cos x, e^{x}, log x, \sqrt{x}, etc. are continuous. Please show your work.

4. (5 points) Assume that f(x) is continuous on the closed interval [0, 1]. Assume also that f(x) ∈ [0, 1] for all x ∈ [0, 1]. Using known theorems about continuous functions, prove that the equation f(x) = x has at least one solution in [0, 1].

5. (3 points each) Which of the following functions are continous and/or uniformly continuous on the specified domain?

(a) 1/x on its natural domain.

(b) 1/x on (1, ∞).

(c) x cos \frac{1}{x} on (0, 10].

(d) \sqrt{x} on [0, ∞).

(e) f(x) = \{ \begin{matrix}|x|/x & for x ≠ 0, \\ 0 & for x = 0, \end{matrix} on (−∞, ∞).

6. (10 points) It is known that the function \sqrt{x} is uniformly continuous on the interval [0.01, 100]. Given ϵ > 0, find a δ > 0 (depending only on ϵ) such that |\sqrt{x} − \sqrt{y}| < ϵ whenever 0.01 ≤ x < y ≤ 100 and |x − y| < δ.