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Problem: Let \( f_{1}=g_{1} \) and \( f_{2}=g_{2} \) a. e. (almost everywhere) on the set \( A \). Prove that \( f_{1}+f_{2}=g_{1}+g_{2} \) a. e. on the set \( A \).

19.6.2.11 Measurable functions and sets

3.25 $

Problem: Let \( f(x) \leq 0 \) a. e. (almost everywhere) on \( A \) and \( g(x) \leq \) \( \leq 0 \) a. e. on \( A \), let \( h(x)=\max (f(x), g(x)) \). Prove that \( h(x) \leq 0 \) a. e. on \( A \).

19.6.2.12 Measurable functions and sets

3 $

Problem: Extend function \( f(x)=x^{3} \) into Fourier series in cosines on the segment line \( [0 ; 1] \).

2.6.2.16 Trigonometric Fourier series

0 $

Problem: Expand function \( f(x)=\sin 2 x \) into Fourier series in cosines on the segment line \( [0 ; 1] \).

2.6.2.17 Trigonometric Fourier series

2 $

Problem: Expand function \( f(x)=\frac{\pi}{4}-\frac{x}{2} \) into Fourier series in cosines on the segment line \( [0 ; 1] \).

2.6.2.18 Trigonometric Fourier series

2 $

Problem: Expand function \( f(x) \) into Fourier series in cosines on the segment line \( [-\pi, \pi] \). Plot the graphs of function \( f(x) \) and sums of Fourier series. Write down the first three partial sums of the Fourier series and plot the graphs: \[ f(x)=\frac{x^{2}-1}{2}, x \in[-\pi, \pi] . \]

2.6.2.19 Trigonometric Fourier series

4.99 $

Problem: Expand periodic function \( f(x) \) with a period of \( T=2 \) into Fourier series. Plot the graphs of function \( f(x) \) and the sums of Fourier series. Write down the first three partial sums of the Fourier series and plot their graphs. \[ f(x)=\left\{\begin{array}{ll} x, & 0 \leq x \leq 1 \\ 1, & 1

2.6.2.20 Trigonometric Fourier series

6.24 $

Problem: 1. Write the first 5 terms of the series: \[ \sum_{n=0}^{\infty} \frac{n^{3}}{n+3} \] 2. Write it as a series with a common term: \[ \frac{0}{2}+\frac{1}{3}+\frac{4}{4}+\frac{9}{5}+\frac{16}{6}+\cdots \]

2.10.33 Number series

0 $

Problem: Examine the convergence of the series and, in the case of convergence, calculate the sum of the series with an accuracy of 0,01 : \[ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sin \frac{1}{n}}{\sqrt{n+2}} \]

2.10.34 Number series

0.75 $

Problem: Calculate the sum: \[ \sum_{k=0}^{n}\left(-4 k^{2}+5 k+2+\frac{4 k+7}{k^{2}+3 k+2}\right) C_{n}^{k} . \]

2.10.35 Number series

4.99 $

Problem: Calculate the sum of the series: \[ \sum_{n=1}^{\infty} \frac{7}{49 n^{2}+35 n-6} . \]

2.10.36 Number series

0.75 $

Problem: Examine the convergence of the series: \[ \sum_{n=1}^{\infty} \frac{n !}{(3 n) !} \text {. } \]

2.10.37 Number series

0 $

Problem: Calculate the sum of the series with an accuracy of \( \alpha=0,1 \) : \[ \sum_{n=0}^{\infty}\left(-\frac{2}{3}\right)^{n} \text {. } \]

2.10.38 Number series

0.75 $

Problem: Examine the convergence of the series and, in the case of convergence, find the sum of the series with an accuracy of 0,01 : \[ \sum_{n=1}^{\infty}(-1)^{n+1} n^{3} \tan ^{5} \frac{\pi}{n} . \]

2.10.39 Number series

1 $

Problem: Examine the convergence of the series and, in the case of convergence, find the sum of the series with an accuracy of 0,01 : \[ \sum_{n=1}^{\infty} \frac{(-1)^{n} \cdot \tan ^{-1} \frac{1}{5 n}}{n !} \]

2.10.40 Number series

1 $

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