Joy of Mathematics

Triangle relation

November 21, 2020 / Leave a comment

Prove that in any triangle $\mathrm{AB} \Gamma$ it holds that

$\left ( \frac{\beta}{\gamma} + \frac{\gamma}{\beta} \right ) \cos \mathrm{A} + \left ( \frac{\gamma}{\alpha} + \frac{\alpha}{\gamma} \right ) \cos \mathrm{B} + \left ( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \right ) \cos \Gamma = 3$

Convergence of series

November 14, 2020 / Leave a comment

Let $\mathrm{gpf}$ denote the greatest prime factor of $n$ . For example $\mathrm{gpf}(17) = 17$ , $\mathrm{gpf} (18) =3$ . Define $\mathrm{gpf}(1)=1$ . Examine if the sum

$\mathcal{S} = \sum_{n=1}^{\infty} \frac{1}{n \; \mathrm{gpf}(n)}$

converges.

On a prime summation

November 13, 2020 / Leave a comment

Let $p_n$ denote the $n$ – th prime. Evaluate the sum

$\mathcal{S} = \sum_{n=1}^{\infty} \frac{ \log p_n}{p_n^2 -1}$

Square summable

November 12, 2020 / Leave a comment

Let $\{a_n\}_{n \in \mathbb{N}}$ be a real sequence such that

If $\{b_n\}_{n \in \mathbb{N}}$ is a real sequence that is square summable; i.e $\sum \limits_{n=1}^{\infty} b_n^2 < +\infty$ the sequence $\sum \limits_{n=1}^{\infty} a_n b_n$ converges.

Prove that $\{a_n\}_{n \in \mathbb{N}}$ is also square summable.

Solution

Let $f_N:\ell_2 \rightarrow \mathbb{R}$ be defined as

$f_N(b)=\sum_{n=1}^{N} a_nb_n$

where $b = (b_n) \in \ell_2$ . We note that

$\begin{align*} |f_N(b)|^2 &\leq \sum_{n=1}^N|a_n|^2 \sum_{n=1}^N|b_n|^2 \\ & \leq ||b||^2\sum_{n=1}^N|a_n|^2 \end{align*}$

Equality holds when $b=(a_1, \dots, a_N, 0, 0, \dots )$ . Hence, $\displaystyle ||f_N||= \left( \sum_{n=1}^N|a_n|^2\right) ^{1/2}$ . From the hypothesis, it follows that $\{f_n\}_{n \in \mathbb{N}}$ is pointwise bounded. It follows from the Uniform boundedness principle ( Banach – Steinhaus ) that $\displaystyle ||f_N||= \left( \sum_{n=1}^N|a_n|^2\right) ^{1/2}$ are bounded. Hence, $\{a_n\}_{n \in \mathbb{N}}$ is square summable.