Power series

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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

1.28 $

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

1.03 $

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

1.28 $

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

1.28 $

Problem: Expand the function \( y(x) \) into a power series in the vicinity of the point \( x_{0} \), where \[ y=\cos ^{2}\left(\frac{\pi x}{6}\right), \quad x_{0}=3 \text {. } \]

2.8.15 Power series

2.57 $

Problem: Calculate the value of the integral with an accuracy of \( \varepsilon=0,01 \), expanding the integrand function into a power series. \[ \int_{0}^{0,5} \frac{d x}{\sqrt[3]{27+x^{3}}} \]

2.8.17 Power series

0.77 $

Problem: Find the area of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{x^{n}}{3 n+8} \text {. } \]

2.8.23 Power series

1.03 $

Problem: Find the area of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{7^{n}(x+3)^{n}}{n} \]

2.8.24 Power series

1.03 $

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

0 $

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

2.8.26 Power series

0 $

Problem: Find the domain of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{n^{3}(x-2)^{3 n+1}}{n^{n+1}} \]

2.8.27 Power series

0.51 $

Problem: Find the domain of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{(x+2)^{n-1}}{\left(2 n^{3}+1\right) 3^{n+1}} \]

2.8.28 Power series

0.77 $

Problem: Find the domain of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2 n}}{(n+1) 5^{n}} \]

2.8.29 Power series

0.77 $

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