Math Problems and solutions
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12.6.1 Higher mathematics
2.55 $
12.6.2 Higher mathematics
3.82 $
12.6.3 Higher mathematics
5.1 $
12.6.4 Higher mathematics
12.6.5 Higher mathematics
6.37 $
12.6.6 Higher mathematics
![Problem: To what and for what \( \alpha \in \mathbb{R} \) does the sequence converge \[ x_{n}=\frac{n 2^{n}+\alpha^{n}}{(n+1) 2^{n}+(2 n+3) \alpha^{n}} ? \]](/storage/uploads/task_mls/1570/en/12.6.1.png){kind=link}
![Problem: Find all functions, differentiable on \( \mathbb{R} \), \( f: \mathbb{R} \rightarrow[1 ;+\infty) \) for which \[ \int_{1}^{f(x)} e^{u^{2}} d u=\int_{0}^{x} \frac{u d u}{f(u)}, \quad \forall x \in \mathbb{R} . \]](/storage/uploads/task_mls/1572/en/12.6.3.png){kind=link}
![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) \( \int_{0}^{1} f(x) d x=\frac{3}{2} \).](/storage/uploads/task_mls/1573/en/12.6.4.png){kind=link}
![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: \[ y_{n}=\frac{1}{1+x_{n}}, \quad S_{n}=\sum_{k=1}^{n} y_{k}, \quad p_{n}=\prod_{k=1}^{n} y_{k} . \]](/storage/uploads/task_mls/1574/en/12.6.5.png){kind=link}
