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Problem: Find the coefficients of the first four non-zero terms of the power series expansion of a particular solution to the differential equation: \( y^{\prime}=x^{2} y^{2}-1, y(0)=1 \).

2.8.2 Power series

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

2.8.3 Power series

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Problem: Calculate the value of the integral \( I \) with an accuracy of \( \varepsilon=0.0001 \), expanding the integrand function into a power series and integrating it term by term: \[ I=\int_{0}^{0.75} \sin \left(-1 x^{2}+3\right) d x \text {. } \]

2.8.4 Power series

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Problem: Find the interval of convergence of the series and examine the convergence at the ends of this interval: \[ \sum_{n=0}^{\infty}(-1)^{n} \cdot \frac{9^{n}(x-1)^{2 n}}{\sqrt[9]{n}-9^{-n}} \]

2.8.5 Power series

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Problem: Expand the function \( f(x) \) into a power series in powers \( (x-a) \), using table expansions. Specify the interval of convergence. \[ f(x)=(x-3) e^{\frac{x}{2},} \quad x_{0}=-3 . \]

2.8.6 Power series

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Problem: Expanding the integrand function into a series, calculate the approximate value of this integral with an error exceeding 0,001 . \[ I=\int_{0}^{1} \frac{d x}{\sqrt[3]{8+x^{2}}} \]

2.8.7 Power series

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Problem: Find a partial solution to the differential equation in the form of a Taylor series under the specified initial conditions (write out four or five terms of the expansion). \[ y^{\prime}=x y-\frac{1}{y}+x,\left.\quad y\right|_{x=-1}=1 . \]

2.8.8 Power series

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Problem: Find the radius and the interval of convergence of the power series, as well as investigate the convergence of the series at the ends of the convergence interval. \[ \sum_{n=0}^{\infty} \frac{(x-1)^{n}}{4^{n}} \]

2.8.9 Power series

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Problem: Calculate the integral \[ \int_{-0.75}^{0} \cos \frac{4 x^{2}}{3} d x \] with an accuracy of 0,001 , expanding the integrand into a power series.

2.8.10 Power series

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Problem: Find the radius of convergence of the given power series and the sum of this series, using theorems on differentiation and integration of series. First include the radius of convergence, and then the sum of the series or the value of the sum at the specified point. \[ \sum_{n=1}^{\infty} n x^{3 n-1}, \quad x_{0}=\frac{1}{2} . \]

2.8.11 Power series

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Problem: Find the domain of convergence of the power series. 1) \( \sum_{n=1}^{\infty} \frac{e^{-n} x^{n}}{\sqrt{n^{3}+1}} \), 2) \( \sum_{n=1}^{\infty} \frac{n^{n}(x+1)^{n}}{n !} \).

2.8.12 Power series

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Problem: Calculate the sum of the series \[ S(x)=\sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{1}{n}-\frac{1}{n+2}\right) x^{n+2} \text {, using } \] term-by-term differentiation and integration series.

2.8.13 Power series

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Problem: Calculate the sum of the series \[ S(x)=\sum_{n=0}^{\infty}\left(n^{2}-2 n-2\right) x^{n} \] using term-by-term differential and integration of power series.

2.8.14 Power series

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Problem: Find the sums of the following series: \[ \sum_{n=1}^{\infty} \frac{x^{n+1}}{\left(1-x^{n}\right)\left(1-x^{n+1}\right)} \] if a) \( |x|<1 \); b) \( |x|>1 \).

2.9.1 Functional sequences and series

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Problem: Find the domain of convergence of the functional series: \[ \sum_{n=1}^{\infty} \frac{2^{n}}{n^{4}} \sin ^{n}(3 x) \text {. } \]

2.9.2 Functional sequences and series

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