Higher mathematics

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Problem: To what and for what \( \alpha \in \mathbb{R} \) does the sequence converge \[ x_{n}=\frac{n 2^{n}+\alpha^{n}}{(n+1) 2^{n}+(2 n+3) \alpha^{n}} ? \]

12.6.1 Higher mathematics

2.41 $

Problem: Prove that the sequence \( n \sin (n) \) has a bounded subsequence. Hint: \( \quad \) use the assertion \( \forall \alpha \in \mathbb{R}, \alpha>0 \), \( \forall M \in \mathbb{N} \exists p, q \in \mathbb{N}, \quad q>M, \quad|\alpha-p / q| \leq 1 / q^{2} \). In other words, any real number is approximated up to \( q^{-2} \) by an infinite set of rational numbers of the form \( p / q \).

12.6.2 Higher mathematics

3.61 $

Problem: Find all functions, differentiable on \( \mathbb{R} \), \( f: \mathbb{R} \rightarrow[1 ;+\infty) \) for which \[ \int_{1}^{f(x)} e^{u^{2}} d u=\int_{0}^{x} \frac{u d u}{f(u)}, \quad \forall x \in \mathbb{R} . \]

12.6.3 Higher mathematics

4.82 $

Problem: Find all such \( f \in C^{2}[0 ; 1] \) functions that 1) \( f(0)=f^{\prime}(0)=1 \), 2) \( f^{\prime \prime}(x) \geq 0, \forall x \in(0 ; 1) \), 3) \( \int_{0}^{1} f(x) d x=\frac{3}{2} \).

12.6.4 Higher mathematics

4.82 $

Problem: A real number \( a \neq-1 \) is given. Let's define the sequence \( x_{n} \) as follows: \( x_{1}=a, x_{n+1}=x_{n}^{2}+x_{n}, n \geq 1 \). Let's also define the sequence \( y_{n} \), as well as the sum and product of its first \( n \) elements: \[ y_{n}=\frac{1}{1+x_{n}}, \quad S_{n}=\sum_{k=1}^{n} y_{k}, \quad p_{n}=\prod_{k=1}^{n} y_{k} . \]

12.6.5 Higher mathematics

6.02 $

Problem: Find the probability that the divisor of \( 10^{99} \) is a multiple of \( 10^{88} \).

12.6.6 Higher mathematics

2.41 $