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Problem: Investigate the functional to an extremum. \[ \begin{array}{l} I[y(x)]=\int_{0}^{1} e^{x}\left(1+y^{2}+\frac{y^{\prime 2}}{2}\right) d x, \\ y(0)=1, \quad y(1)=e \end{array} \]

4.15 Variational calculus

3 $

Problem: For the Bolza problem find the admissible extremal and determine its type: \[ B=\int_{1}^{2} t^{2} \dot{x}^{2} d t-2 x(1)+x^{2}(2) \rightarrow \text { extr. } \]

4.16 Variational calculus

3.75 $

Problem: Find the eigenvalues and the eigenfunctions of the functional: \[ \begin{array}{l} \mathcal{T}[y]=\int_{0}^{1}\left(y^{2}+\left(y^{\prime}\right)^{2}\right) d x, \quad y(0)=y(1)=0, \\ \int_{0}^{1} y^{2} d x=0 . \end{array} \]

4.17 Variational calculus

3.75 $

Problem: Solve the Bolza problem: \[ B(x(\cdot))=\int_{1}^{2} t^{2} \dot{x}^{2} d t-2 x(1)+x^{2}(2) \rightarrow \text { extr. } \]

4.18 Variational calculus

4.25 $

Problem: For the simplest problem of the classical calculus of variations describe the set of admissible functions and investigate the functional for an extremum. \[ \begin{array}{l} \mathrm{J}=\int_{1}^{2}\left(\dot{x}^{2}-t^{2} x\right) d t \rightarrow \text { extr }, \\ x(0)=0, \quad x(1)=0 . \end{array} \]

4.19 Variational calculus

4.25 $

Problem: With an accuracy of \( \varepsilon=0,001 \) find the partial sum \( S \) of the series \[ S=\sum_{n=1}^{\infty} \frac{(3 n+2)^{2}}{7^{n}} . \]

2.10.4 Number series

3.75 $

Problem: With an accuracy of \( \varepsilon=0,001 \) find the partial sum \( S \) of the series \[ S=\sum_{n=1}^{\infty} \frac{n}{(3 n+2)^{2}} \text {. } \]

2.10.5 Number series

3.75 $

Problem: Examine the series for the absolute and conditional convergence: \[ \sum_{n=1}^{\infty}(-1)^{n+1} \cdot \frac{1}{3 n-1} \text {. } \]

2.10.6 Number series

0.5 $

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=1}^{\infty}(-1)^{n+1} \cdot a_{n}, \quad \text { where } a_{n}=\frac{1 \cdot 4 \cdot \ldots \cdot(3 n-2)}{3 \cdot 5 \cdot \ldots \cdot(2 n+1)} \]

2.10.7 Number series

0.75 $

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=1}^{\infty}(-1)^{n} \cdot \frac{\ln n}{n} . \]

2.10.8 Number series

0.75 $

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=3}^{\infty}(-1)^{n} \cdot \frac{1}{n \ln n(\ln \ln n)^{3}} . \]

2.10.9 Number series

1.25 $

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=1}^{\infty}(-1)^{n-1} \cdot \frac{n^{2}}{2^{n}} . \]

2.10.10 Number series

0.75 $

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=3}^{\infty}(-1)^{n} \cdot \frac{1}{n \ln n \sqrt{\ln \ln n}} \]

2.10.11 Number series

1.25 $

Problem: Make sure that Leibniz test can't be applied to the series \( \sum_{n=1}^{\infty} N_{n} \) with the terms given below \( (k \in N) \). Examine this series for convergence in other ways. \[ a_{n}=\left\{\begin{array}{lc} \frac{1}{\sqrt{k+1}+1}, \quad n=2 k-1 \\ -\frac{1}{\sqrt{k+1}-1}, \quad n=2 k \end{array}\right. \]

2.10.12 Number series

1.25 $

Problem: Make sure that the Leibniz test cannot be applied to the series \( \sum_{n=1}^{\infty} N_{n} \) with the terms \( (k \in N) \) indicated below. Examine this series for convergence in other ways. \[ a_{n}=\left\{\begin{array}{cc} \frac{1}{3^{k}}, & n=2 k-1 \\ -\frac{1}{2^{k}}, \quad n=2 k \end{array}\right. \]

2.10.13 Number series

1.25 $

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