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Problem: Examine the series for convergence: 1) \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}(n+1)} \), 2) \( \sum_{n=1}^{\infty} \frac{n^{4}}{4^{n}+1} \), 3) \( \sum_{n=1}^{\infty}\left(\frac{n}{3 n-1}\right)^{2 n-1} \).

2.10.29 Number series

2.08 $

Problem: Calculate the sum of the series: \[ \sum_{n=1}^{\infty} \frac{12}{36 n^{2}-12 n-35} . \]

2.10.30 Number series

1.04 $

Problem: Calculate the sum of the series: \[ \sum_{n=1}^{\infty} \frac{3}{n(n-1)(n+3)} \]

2.10.31 Number series

1.3 $

Problem: Examine the series for convergence: \[ \sum_{n=1}^{\infty} \frac{1}{n \sqrt{(\ln n+2)^{5}}} . \]

2.10.32 Number series

1.3 $

Problem: Expand function \( y(x) \) into a power series in the vicinity of the point \( x_{0} \), where 1) \( y=9^{3 x}, \quad x_{0}=-2 \), 2) \( y=\frac{1}{x^{2}+9 x+20}, \quad x_{0}=0 \).

2.8.16 Power series

2.6 $

Problem: Calculate the value of the integral with an accuracy of \( \varepsilon=0,01 \), expanding the integrand function into a power series. \[ \int_{0}^{0,1} \sin 8 x^{2} d x \text {. } \]

2.8.22 Power series

1.3 $

Problem: Find the domain of convergence of the power series. \[ \sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{n+2} \]

2.8.18 Power series

1.04 $

Problem: Applying the Taylor formula with a remainder term in Lagrange form to function \( f(x)=e^{x} \), calculate the value of \( e^{a} \) with an accuracy of 0,001. \( a=0,33 \).

2.8.19 Power series

1.3 $

Problem: Expand function \( f(x)=e^{2 x} \) into a power series in the vicinity of the point \( x_{0}=3 \).

2.8.20 Power series

0 $

Problem: Expand function \( y(x) \) into a power series in the vicinity of point \( x_{0} \), when \[ y=\ln \frac{2+x}{3+4 x}, x_{0}=0 \text {. } \]

2.8.21 Power series

1.3 $

Problem: Do the points \( A(1 ; \Gamma ; 2), B(-1 ; H ; 3), C(0 ; \Gamma ; 4) \), \( D(2 ; 2 ; \mathrm{H}) \) lie on the same plane? \[ \Gamma=\mathrm{H}=2 \text {. } \]

3.3.1 Line and plane in space

0 $

Problem: Find intersection points of the surface \( \Gamma x y+x^{2}-y^{2}=z \) and the straight line \[ \frac{x+\Gamma}{-1}=\frac{y-\mathrm{H}}{1}=\frac{z+\mathrm{H}}{-2} \text {, where } \Gamma=\mathrm{H}=2 \text {. } \]

3.3.2 Line and plane in space

0.52 $

Problem: Create a truth table for function \( f=a \& \bar{b} \oplus(b \bigvee c) \). Using the table, write the PDNF and PCNF.

6.3.3 Boolean algebra

2.6 $

Problem: Determine which vertices are the source or sink of a given graph. Find strongly connected components and construct a factor graph. Give the necessary explanations.

6.4.1 Graph theory

1.3 $

Problem: Find number of perfect matching. a) in the complete graph \( K_{5} \), b) in the complete bipartite graph \( K_{5,5} \), c) in the graph obtained from the complete bipartite graph \( K_{5,5} \) by erasing four edges adjacent to the vertex \( V_{6} \).

6.4.2 Graph theory

5.21 $

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