Chapter 1: Infinite Series, Power Series

Use the integral test to show that$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}{\mathit{e}}^{\mathbf{-}{\mathbf{n}}^{\mathbf{2}}}$ converges.

Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point.

$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}}}{\mathbf{\left(}\mathbf{2}\mathbf{n}\mathbf{\right)}\mathbf{!}}{\left(\frac{\pi }{2}\right)}^{\mathbf{2}\mathbf{n}}$

Find the Maclaurin series of the following functions.

$\mathbf{arc}\mathit{t}\mathit{a}\mathit{n}\mathit{x}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{x}}\frac{\mathbf{d}\mathbf{u}}{\mathbf{1}\mathbf{+}{\mathbf{u}}^{\mathbf{2}}}$

Use the ratio test to find whether the following series converge or diverge:

18.$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{2}}^{\mathbf{n}}}{{\mathbf{n}}^{\mathbf{2}}}$

Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{1}}{\mathbf{n}{\mathbf{2}}^{\mathbf{n}}}$.

Find the geodesics on the cone ${\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{=}{\mathit{z}}^{\mathbf{2}}$. Hint: Use cylindrical coordinates.

Find eigenvalues and eigenvectors of the matrices in the following problems.

$\left(\begin{array}{cc}5& 1\\ 4& 2\end{array}\right)$

Find the few terms of the Taylor series for the following function about the given points.

$\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathbf{,}\mathit{a}\mathbf{=}\mathit{\pi }$

Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\left(\begin{array}{c}-\frac{1}{2}\\ n\end{array}\right){\left(-\frac{1}{2}\right)}^{\mathbf{n}}$.

Use the ratio test to find whether the following series converge or diverge:

19.$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{3}}^{\mathbf{n}}}{{\mathbf{2}}^{\mathbf{2}\mathbf{n}}}$