Chapter 1: Infinite Series, Power Series

Use power series to evaluate the function at the given point. Compare with computer results, using the computer to find the series, and also to do the problem without series. Resolve any disagreement in results (see Example 1).${\mathbf{e}}^{\mathbf{arcsin}\mathbf{}\mathbf{x}}\mathbf{+}\mathbf{In}\mathbf{\left(}\frac{\mathbf{1}\mathbf{-}\mathbf{x}}{\mathbf{e}}\mathbf{\right)}\mathbf{at}\mathbf{}\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0003}$.

Solve the equations (9.3) for x and y in terms of u and v Use your equations to sketch the images in the w plane of the z plane lines $\mathbf{x}\mathbf{=}\mathbf{const}\mathbf{.}$ (for several values of x ) and similarly of$\mathbf{y}\mathbf{=}\mathbf{const}\mathbf{.}$

Question:Find the steady-state temperature distribution inside a sphere of radius 1 when the surface temperatures are as given in Problems 1 to 10.
$\mathbf{35}{\mathbf{cos}}^{\mathbf{4}}\mathbf{\theta }$.

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

$\sqrt[\mathbf{3}]{\mathbf{x}}\mathbf{,}\mathit{a}\mathbf{=}\mathbf{8}$

$e^{x}sinx$

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

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

By computer or tables, find the exact sum of each of the following series.

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

Solve the differential equations in Problems 5 to 10 by the Frobenius method; observe that you get only one solution. (Note, also, that the two values of are equal or differ by an integer, and in the latter case the larger gives the one solution.) Show that the conditions of Fuchs's theorem are satisfied. Knowing that the second solution is X times the solution you have, plus another Frobenius series, find the second solution.

$\mathit{x}{\mathbf{(}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathit{y}\mathbf{"}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{=}\mathbf{0}$

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

${\mathit{e}}^{\mathbf{x}}\mathbf{,}\mathit{a}\mathbf{=}\mathbf{1}$

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

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