A double Putnam 2016 series

Evaluate the series:

$$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}$$

(Putnam 2016)

Solution

Not Lebesgue integrable function

Let $x \in \mathbb{R}$. Given the series:

\begin{equation} \sum_{n=2}^{\infty} \frac{\sin nx}{\ln n} \end{equation}

1. Prove that $(1)$ converges forall $x \in \mathbb{R}$.
2. Prove that $(1)$ is not a Fourier series of a Lebesgue integrable function.

Solution

$\mathbb{R}^2 \rightarrow \mathbb{R}$

Prove that there does not exist an $1-1$ and continuous mapping from $\mathbb{R}^2$ to $\mathbb{R}$.

Solution

Existence of constant

Let $f:[0, 1] \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=0$ and

$$\int_0^1 f(x) \, {\rm d}x = \int_0^1 x f(x) \, {\rm d}x \tag{1}$$

Prove that there exists a $c \in (0, 1)$ such that

$$\int_0^c x f(x) \, {\rm d}x = \frac{c}{2} \int_0^c f(x) \, {\rm d}x$$

Solution

Convergence and dyadic numbers

A real number $x$ is said to be dyadic rational provided there is an integer $k$ and a non negative integer $n$ for which $\displaystyle x=\frac{k}{2^n}$ . For each $x \in [0, 1]$ and each $n \in \mathbb{N}$ set:

$$f_n(x) = \left\{\begin{matrix} 1 &, & x =\dfrac{k}{2^n} , \; k \in \mathbb{N} \\ 0& , & \text{otherwise} \end{matrix}\right.$$

1. Prove that the dyadic numbers are dense in $\mathbb{R}$.
2. Let $f:[0, 1] \rightarrow \mathbb{R}$ be the function to which the sequence $\{f_n\}_{n \in \mathbb{N}}$ converges pointwise. Prove that $\bigintsss_0^1 f(x) \, {\rm d}x$ does not exist.
3. Show that the convergence $f_n \rightarrow f$ is not uniform.

Solution

An evaluation of integral with unknown $f$

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function in $[0, 1]$, strictly monotonic and $f(0)=1$ . If forall $x \in \mathbb{R}$ holds $f\left( f(x) \right)=x$ then evaluate the integral

$$\mathcal{J} = \int_0^1 \left(x - f(x) \right)^{2016} \, {\rm d}x$$

Solution

Integral and inequality

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a positive real valued and continuous function such that it is periodic of period $T=1$. Prove that:

$$\int_0^1 \frac{f(x)}{f \left(x + \frac{1}{2} \right)}\, {\rm d}x  \geq 1$$

Solution

Probably a beauty and a beast

Here is something that arose while playing around with a class of particular integrals.

Let $\displaystyle a_n= \int_0^1 \left( e^x - 1 - x - \frac{x^2}{2} - \cdots - \frac{x^n}{n!} \right) \, {\rm d}x$. Evaluate the series

$$\mathcal{S}=\sum_{n=0}^{\infty} a_n$$

Solution

On functions

Here are some examples on functions with "strange" properties.

Give an example of a function that:

1. is only continuous at a single point.
2. is continuous at isolated points.

Does there exist a function that:

1. is continuous only on the rationals?
2. is continuous only on the irrationals?

Explain your answer giving a brief explanation. 

Solution

Double alternating sum

Let $\displaystyle c_n= \sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k+n}$. Evaluate the sum:

$$\mathcal{S}=\sum_{n=1}^{\infty} \frac{c_n}{n}$$

Solution

A beautiful double sum

Prove that:

$$ \sum_{n=1}^{\infty}\left [ \frac{1}{n} \sum_{m=0}^{\infty} \frac{1}{(2m+1)^{2n}} - \log \left ( 1 + \frac{1}{n} \right ) \right ]$$

Solution

On a known inequality

Let $x$ be a real number and let $n \in \mathbb{N}$. Prove the inequality:

$$\left | \sum_{k=1}^{n} \frac{\sin kx}{k} \right |\leq 2 \sqrt{\pi}$$

Solution

Zero function

The following is an exercise proposed by one of our readers.

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f\left( \frac{m}{2^n} \right)=0 \; \text{forall} \; m \in \mathbb{Z} \; \text{and} \; n \in \mathbb{N}$. Prove that $f=0$ forall $x \in \mathbb{R}$.

Solution

An equality of two finite sums

Let $n \in \mathbb{N}$. Prove that:

$$\sum_{k=1}^{n}\frac{1}{4k^{2}-2k}=\sum_{k=n+1}^{2n}\frac{1}{k}$$

Solution

A limit with floor function

Let $\left \lfloor \cdot \right \rfloor$ denote the floor function. Prove that:

$$\lim_{x\rightarrow 0} \; x \left \lfloor \frac{1}{x} \right \rfloor=1$$

Solution

Example of a function

Give an example of a continuous function $f:[1, +\infty) \rightarrow \mathbb{R}$ such that $f(x)>0$ forall $x \in [1, +\infty)$ and $\bigintsss_{1}^{\infty} f(x) \, {\rm d}x$ converges while $\bigintsss_{1}^{\infty} f^2(x) \, {\rm d}x$ diverges.

Solution

Constant as periodic

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a periodic function such that $\lim \limits_{x \rightarrow +\infty} f(x) =\ell \in \mathbb{R}$. Prove that $f$ is constant.

Solution

$\mathbb{K}_n$ series

The $\mathbb{K}_n$ series ( or Kempner series) are obtained by removing all terms containing a single digit $d$ from the harmonic series. For example:

$$\mathbb{K}_1= \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{20}+ \frac{1}{22}+ \frac{1}{23}+\cdots$$

All $\mathbb{K}$ ten series converge as strangly as it sounds. In this topic we shall prove the convergence of $\mathbb{K}_1$.

Solution

Constant function as periodic

Prove that every continuous and periodic function that has every $\frac{1}{n}$ as a period is constant.

Solution

Absolute convergence of a series

Let $y_n , \; n \in \mathbb{N}$ be a sequence of real numbers such that for all real valued sequences $x_n , \; n \in \mathbb{N}$ such that $\lim x_n =0$ the series $\sum \limits_{n=1}^{\infty} x_n y_n $ converges. Does it necessarily follow that the series $\sum \limits_{n=1}^{\infty} \left| y_n \right|$ converges?

Solution