Chapter 1: Infinite Series, Power Series

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

21.$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{5}}^{\mathbf{n}}{\left(n!\right)}^{\mathbf{2}}}{\left(2n\right)\mathbf{!}}$

Find the interior temperature in a hemisphere if the curved surface is held at $\mathit{u}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }$and the equatorial plane at $\mathit{u}\mathbf{=}\mathbf{1}$.

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

The series $\mathbf{\sum }_{\mathbf{n}\mathbf{-}\mathbf{1}}^{\mathbf{\infty }}\mathbf{1}\mathbf{/}{\mathit{n}}^{\mathbf{s}}$,$\mathit{s}\mathbf{>}\mathbf{1}\mathbf{,}$is called the Riemann Zeta function,$\mathit{\zeta }\left(s\right)$. (In Problem$\mathbf{14}\mathbf{.}\mathbf{2}\left(a\right)$you found$\mathit{\zeta }\left(2\right)\mathbf{=}\frac{{\mathbf{\pi }}^{\mathbf{2}}}{\mathbf{6}}$. When$\mathit{n}$is an even integer, these series can be summed exactly in terms of$\mathbf{\pi }$.) By computer or tables, find

$$\begin{gathered} \left(a\right)\mathit{\zeta }\left(4\right)\mathbf{=}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{4}}} \\ \left(b\right)\mathit{\zeta }\left(3\right)\mathbf{=}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{3}}} \\ \left(c\right)\mathit{\zeta }\left(\frac{3}{2}\right)\mathbf{=}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{3}\mathbf{/}\mathbf{2}}} \end{gathered}$$

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

22.$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{10}}^{\mathbf{n}}}{{\left(n!\right)}^{\mathbf{2}}}$

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

Use the series you know to show that:

$\frac{{\mathbf{\pi }}^{\mathbf{2}}}{\mathbf{3}\mathbf{!}}\mathbf{-}\frac{{\mathbf{\pi }}^{\mathbf{4}}}{\mathbf{5}\mathbf{!}}\mathbf{+}\frac{{\mathbf{\pi }}^{\mathbf{6}}}{\mathbf{7}\mathbf{!}}\mathbf{-}\mathbf{\dots }\mathbf{=}\mathbf{1}$

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

23.$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{n}\mathbf{!}}{{\mathbf{100}}^{\mathbf{n}}}$

Find the following limits using Maclaurin series and check your results by computer. Hint: First combine the fractions. Then find the first term of the denominator series and the first term of the numerator series.

$$\begin{gathered} \left(a\right)\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{}\mathbf{lim}}\left(\frac{1}{x}-\frac{1}{e^x-1}\right) \\ \left(b\right)\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\left(\frac{1}{x^2}-\frac{\cos x}{{\sin }^{2}x}\right) \\ \left(c\right)\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\left({\csc }^{2}x-\frac{1}{x^2}\right) \\ \left(d\right)\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\left(\frac{\mathrm{In}\left(1+x\right)}{x^2}-\frac{1}{x}\right) \end{gathered}$$

Find the following limits using Maclaurin series and check your results by computer. Hint: First combine the fractions. Then find the first term of the denominator series and the first term of the numerator series.

$\mathit{a}\mathbf{)}\mathbf{}\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{l}\mathbf{i}\mathbf{m}}\left(\frac{1}{x}-\frac{1}{e^x-1}\right)$

role="math" localid="1662640763230" $\mathit{b}\mathbf{)}\mathbf{\hspace{0.17em}}\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{l}\mathbf{i}\mathbf{m}}\mathbf{(}\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{-}\frac{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{x}}{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{x}}\mathbf{)}$

$\mathit{c}\mathbf{)}\mathbf{}\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\left(cs{c}^{2}x-\frac{1}{x^2}\right)$