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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.