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Here is a solution. Let \( \mathcal{H}_k \) denote the \(k \) - th harmonic number. We are evaluating the limit according to the limit derivative definition: $$\lim_{x\rightarrow 0}\frac{f(x)-f(0)}{x}= f'(0)$$ So , according to the above limit we split the limit accordingly: \( \begin{align*} \lim_{...

Let \( \mathbf{a, b, c} \in \mathbb{R}^3 \) be three vectors. Solve the equation:

$$\mathbf{x+(x\cdot a)b = c}$$

Let \( \Gamma \) be Euler's Gamma function. Prove that:

$$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > 0,\; 0 < s < 1$$

This inequality is better known as Gautchi's Inequality and the proof is not that hard.

Demetres wrote: Tolis, please confirm if the answer is $45289$.

Affirmative.

As for the evil mind , no idea.

Thank you Vaggelis. Ramanujan also proved that: $$\begin{equation} \text{Ti}_2 (y)-\text{Ti}_2 \left(\frac{1}{y} \right)=\frac{\pi}{2}\log y, \quad y>0 \end{equation}$$ Setting \( 2-\sqrt{3} \) at \( (1) \) we get that: $$\text{Ti}_2 \left( 2+\sqrt{3}\right) = \frac{5\pi }{12} \log \left( 2+\sqrt{3}...

Let \( \displaystyle \text{Ti}_2(x) = \int_0^x \frac{\arctan t}{t}\, {\rm d}t \) . Ramanujan proved that: $$\text{Ti}_2 \left(2-\sqrt{3} \right)=\frac{\pi}{12}\log \left(2-\sqrt{3} \right) +\frac{2}{3}\mathcal{G}$$ where \( \mathcal{G} \) is the Catalan's constant . Can you provide a proof for the r...

Give an example of a function \( f:\mathbb{R} \rightarrow \mathbb{R} \) such that any rational number is its period but any irrational is not. Also, prove that there exists no function \( g:\mathbb{R} \rightarrow \mathbb{R} \) such that any irrational is its period and any rational is not.

Let \( A, B \in \mathbb{R}^{n \times n}\) that are diagonizable in \( \mathbb{R}\) . If \( \det (A^2+B^2)=0\) and \( AB=BA \) , then prove that \( \det A = \det B =0\) .

Let \( a_n \) be a strictly increasing sequence of positive integers. Prove that:

$$x_n =\frac{1}{a_1} + \frac{1}{a_1 a_2} +\cdots + \frac{1}{a_1 a_2 a_3 \cdots a_n}$$

converges to an irrational number.

Evaluate the limit:

$$\lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{k \leq n} \varphi(k)$$

where $\varphi$ is the Euler's function.