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Here is something I created.

Let $\varphi$ denote Euler’s totient function. Evaluate the limit

$$\ell = \lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{k=1}^{n} \sin \left (\frac{\pi k}{n} \right) \varphi(k)$$

Let us denote with $\zeta$ the Riemann zeta function defined as $\zeta(0)=-\frac{1}{2}$. Let us also denote with $\zeta^{(n)}$ the $n$-th derivative of zeta. Evaluate the limit

$$\ell=\lim_{n \rightarrow +\infty} \frac{\zeta^{(n)}(0)}{n!}$$

We show that the series $\sum_{n=1}^\infty\,f(n)/n^2$ diverges for any bijective from $\mathbb{N}$ to $\mathbb{N}$. To see this, since $f$ is a permutation of $\mathbb{N}$, it follows that $$f(1) + f(2) + \cdots + f(n) \geq 1 + 2 + \cdots + n = \frac{n(n+1)}{2}.$$ Thus, by Abel's summation formula,...

Greetings, I have assembled here some problems in Analysis that are quite entertaining and some of them are quite challenging. The following booklet contains problems from $4$ main branches of Mathematical Analysis: Real and Complex Analysis Multivariable Calculus General Topology Integrals and Seri...

Prove that the number

$$\mathcal{N}=(2+\sqrt{3})^n +(2-\sqrt{3})^n$$

is an integer.

Let us denote the $n$ -th Lucas number as $L_n$. It is known that $L_0=2$ , $L_1=1$ as well as

$$L_{n+2}=L_{n+1} + L_{n} \quad \text{forall} \; n \geq 0$$

Calculate the value of

$$\mathcal{R}=\sqrt[n]{L_{2n} - \frac{1}{L_{2n}- \dfrac{1}{L_{2n}-\ddots}}}$$

I don't have a solution.

Let

$$\mathbb{S}=\left\{ (x, y, z) \in \mathbb{R}^3 \big| x^2 + y^2 +z^2 \leq 1 \right\}$$

Evaluate the surface integral:

$$\mathfrak{S}=\iiint \limits_{\mathbb{S}} \cosh (x + y + z ) \, {\rm d} (x, y, z)$$

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

$$\mathfrak{n}=\sqrt{\underbrace{1111\cdots11}_{2n} - \underbrace{2222\cdots22}_{n}}$$

is rational.

Let $a,b, c$ be positive real numbers. Prove that:

$$\sqrt{\frac{2a}{a+b}} +\sqrt{\frac{2b}{b+c}} + \sqrt{\frac{2c}{c+a}} \leq 3$$

(V. Cirtoaje)

Here is an exercise that caught my attention the other day. Let $\mathcal{G}$ be a finite group such that $\left ( \left | \mathcal{G} \right | , 3 \right ) =1$. If for the elements $a, \beta \in \mathcal{G}$ holds that: $$\left ( a \beta \right )^3 = a^3 \beta^3$$ then prove that $\mathcal{G}$ is ...