OSA: Real Analysis

Question 1

Which of the following statements are correct?

statementcorrect?resultexplanation

Every sequence $$\{x_n\}_{n \geq 1}$$ of real numbers that satisfies $$|x_n - x_{2n}| \to 0$$ is convergent.

For every sequence $$\{x_n\}_{n \geq 1}$$ of real numbers the sequence $$\{y_n\}_{n \geq 1}$$ with $y_n := \frac{1}{1+x_n^2}$ has a convergent subsequence.

For every set $$A \subseteq \mathbb{R}$$ we let $$\exp(A) := \{ e^a | a \in A \}$$. Then $$\exp(A)$$ has a finite infimum for every nonempty $$A$$.

A set is closed if, and only if, it is not open.

Let $$K_n \subseteq \mathbb{R}$$ be compact for every $$n \geq 1$$. Then the intersection $$\bigcap_{n \geq 1} K_n$$ is compact as well.

Let $$f:(0,\infty)\to \mathbb{R}$$ be continuous. Then $x \mapsto \frac{1}{1+f(x)^2}$ has a limit for $$x \to 0+$$ (i.e. $$x$$ tending to $$0$$ from the right).

Let $$f : \mathbb{R}\to\mathbb{R}$$ be twice continuously differentiable. Then the function $$x \mapsto |f(x)|$$ is continuously differentiable on $$\mathbb{R}$$.

Let $$\{f_n\}_{n \geq 1}$$ be a uniformly convergent sequence of real functions on $$[0,1]$$. Then the sequence $$\{|f_n|\}_{n \geq 1}$$ is uniformly convergent as well.

Question 2

Check the following series for convergence and determine its limit, if it exists: $\sum_{n=1}^\infty \frac{n^2+3n}{n^3+n^2-n+5}$

Determine the set of $$x \in \mathbf{R}$$, for which $\sum_{k=1}^\infty \frac{(k!)^2}{(3k)!}(x-5)^2k$ converges.

Note: Since this question requires you to provide a rigorous justification of your claims (i.e. a proof), your answer can not be checked automatically. We invite you to send your solutions to this email address and will get back to you with feedback.

Question 3

Let the function $$f_n:[0,\infty)\to \mathbf{R}$$ for $$n \geq 1$$ be defined as $f_n(x) := \int_0^x e^{-\frac{t^2}{n}}\,\mathrm{d}t.$

1. Show that $$f_n$$ is continuously differentiable on $$(0,\infty)$$ for each $$n$$.
2. Show that for every $$x \geq 0$$ the limit $$\lim_{n \to \infty} f_n(x)$$ exists and determine its value.

Note: Since this question requires you to provide a rigorous justification of your claims (i.e. a proof), your answer can not be checked automatically. We invite you to send your solutions to this email address and will get back to you with feedback.