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Problem: Solve for all values of the parameter: \[ a^{2}(x-1)-a(2 x-5)=3 x-6 \text {. } \]

17.26 USE problems

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Problem: Solve for all values of the parameter. \[ 3 x+9

17.27 USE problems

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Problem: Solve for all values of the parameter. \[ \frac{x-5}{x-1}-\frac{2}{k}=\frac{3}{k(x-1)} \text {. } \]

17.28 USE problems

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Problem: Find all values of the parameter for which the equation \( a x^{2}-4 a x+4 a-5=0 \) has negative roots.

17.29 USE problems

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Problem: Solve the irrational equation. \[ \sqrt{3 x^{2}+5 x+8}-\sqrt{3 x^{2}+5 x+5}=1 . \]

17.30 USE problems

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Problem: Find the coordinates of the center and the radius of the circle: \[ x^{2}+y^{2}-8 x+6 y=0 \text {. } \] 밤 \( (\mathrm{Ctrl}) \cdot \)

3.1.5 Curves of the 2-nd order

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Problem: Find the domain of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{n x^{n}}{2^{n}} \text {. } \]

2.8.25 Power series

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

2.8.26 Power series

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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

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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

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

2.10.37 Number series

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Problem: Examine the convergence of the series: \[ \sum_{n=0}^{\infty} \frac{2 n+n^{5}}{3 n^{7}-7} . \]

2.10.41 Number series

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Problem: Examine the convergence of the series: \[ \sum_{n=0}^{\infty}\left(\frac{3}{4 n+7}\right)^{n} \text {. } \]

2.10.43 Number series

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Problem: Determine the type of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(n+2)^{n}} \text {. } \]

2.10.44 Number series

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Problem: Determine the type of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{(-1)^{n} 6 n}{2 n} \text {. } \]

2.10.45 Number series

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