Math 5852 Problems

Fall 2021

  1. (Due 9/10/21) Show that a sequence of functions \( \{f_n\} \) fails to converge to a function \(f\) uniformly on a set \(E\) if and only if there is some positive \(\varepsilon_0\) so that a sequence \(\{x_k\}\) of points in \(E\) and a subsequence \(\{f_{n_k}\}\) can be found such that $$|f_{n_k}(x_k) - f(x_k)| \ge \varepsilon_0.$$ (9.3.13, Thomson & Bruckner)


  2. (Due 9/15/21) Show that \(\left\{\frac{nx}{1+nx} \right\}_{n=1}^\infty\) converges uniformly on \([a, \infty)\) for any \(a > 0.\) \( \ \) Does this sequence converge uniformly on \([0, \infty)\)? \( \ \) Explain.


  3. (Due 9/20/21) Compute the limit $$\lim_{n\to \infty} \int_0^1 \frac{e^{-nt}}{\sqrt t} \ dt.$$ (9.5.10, Thomson & Bruckner)


  4. (Due 10/4/21) Find the sum of \(\displaystyle \sum_{n=1}^\infty \frac{1}{n(n+2) 2^n}\). \( \ \)(Hint: Consider the function \(\displaystyle f(x) = \sum_{n=1}^\infty \frac{x^n}{n(n+2)}\).)


  5. (Due 10/13/21) Express \(\displaystyle \int_0^1 x^{-x} dx\) as a sum \(\displaystyle \sum_{n=0}^\infty a_n. \ \) (Hint: \(x^{-x} = e^{-x \ln x}. \ \) Use the power series for \(e^x.\))


  6. (Due 10/20/21) Find \(m_1\), \(m_2\), \(m_3\), and \(m_4\) such that for all \(x \in \mathbb{R}^n\) $$m_1 \|x\|_1 \le \|x\| \le m_2 \|x\|_1$$ and $$m_3 \|x\|_\infty \le \|x\| \le m_4 \|x\|_\infty$$ and such that these \(m_i\)'s are best possible.


  7. (Due 10/27/21) Let \(E\) be a bounded non-empty set in \(\mathbb{R}^n. \ \) Show that $$\text{diam}(E) = \text{diam}(\text{bd}(E)).$$


  8. (Due 11/17/21) Let \(\ f : \mathbb{R}^2 \to \mathbb{R}^2\ \) be defined by \(f(x, y) = (x^2-y^2, 2xy). \ \) Show that \(f\) is continuous.


  9. (Due 11/29/21) Evaluate $$\int_0^1 \frac{x - 1}{\ln x} \, dx.$$ Hint: Consider \(\int_0^1 \, \frac{x^p-1}{\ln x} \, dx. \)


  10. (Due 12/3/21) Evaluate $$\int_0^1 \frac{\ln(x + 1)}{x^2+1} \, dx.$$ Hint: Consider \(\int_0^1 \, \frac{\ln(xy + 1)}{x^2 + 1} \, dx. \)