Let $$f(x) = \sum_{n \geq 1} \frac{\sin\left(\frac{x}{n}\right)}{n}$$ Show that $f$ is continuous. Determine (with justification) whether $f$ is differentiable.

For a strictly positive rational number $a \in \mathbf{Q}_{>0}$, let $\log(a)$ denote the unique real number satisfying $e^{\log a} = a$. Deduce from Theorem 1 that $\log(a)$ is irrational for every strictly positive rational number $a \neq 1$.
(Theorem 1: Let $r \geq 2$ be an integer. If $a_1, \ldots, a_r \in \mathbf{Q}$ are distinct rational numbers, then the real numbers $e^{a_1}, \ldots, e^{a_r}$ are linearly independent over $\mathbf{Q}$.)

Let $\alpha \in \mathbb { R } _ { + }$. We assume that there exist two sequences of non-zero natural integers $\left( p _ { n } \right) _ { n \in \mathbb { N } }$ and $\left( q _ { n } \right) _ { n \in \mathbb { N } }$ such that
$$\lim _ { n \rightarrow + \infty } \frac { p _ { n } } { q _ { n } } = \alpha \quad \text { and } \quad \left| \alpha - \frac { p _ { n } } { q _ { n } } \right| \underset { n \rightarrow + \infty } { = } o \left( \frac { 1 } { q _ { n } } \right)$$
We further assume that for all $n \in \mathbb { N } , \frac { p _ { n } } { q _ { n } } \neq \alpha$.
Show that $\alpha$ is an irrational number.

Let $\beta = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { 10 ^ { n ! } }$.
Justify that $\beta$ is well-defined, then show that $\beta$ is an irrational number.

Let $n \in \mathbb { N } ^ { * }$. Justify that $\zeta ( 2 ) = \sum _ { k = 1 } ^ { + \infty } \frac { 1 } { k ^ { 2 } }$ is well-defined, then show that we can write
$$\sum _ { k = 1 } ^ { n } \frac { 1 } { k ^ { 2 } } = \frac { p _ { n } } { q _ { n } }$$
with $p _ { n } \in \mathbb { N } ^ { * }$ and $q _ { n } = d _ { n } ^ { 2 }$.

Can we apply the result of Q20 to these sequences $\left( p _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ and $\left( q _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ to conclude about the irrationality of $\zeta ( 2 )$ ?

Let $x$ be an irrational number. If $a , b , c$ and $d$ are rational numbers such that $\frac { a x + b } { c x + d }$ is a rational number, which of the following must be true?
(A) $a d = b c$
(B) $a c = b d$.
(C) $a b = c d$.
(D) $a = d = 0$