# 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 $$\int_0^1 \, \frac{\ln(xy + 1)}{x^2 + 1} \, dx.$$