Chapter 1: Infinite Series, Power Series

An aquarium with rectangular sides and bottom (and no top) is to hold 5gel. Find its proportions so that it will use the least amount of material.

Show that the interval of convergence of the series$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{x}}^{\mathbf{n}}}{{\mathbf{n}}^{\mathbf{2}}\mathbf{+}\mathbf{n}}$is $\left|x\right|\mathbf{\le }\mathbf{1}\mathbf{}\mathbf{(}\mathbf{For}\mathbf{}\mathbf{x}\mathbf{=}\mathbf{1}$. (For , this is the series of Problem 9.) Using theorem$\left(14.4\right)$, show that for$\mathit{x}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$, four terms will give two decimal place accuracy.

Write and solve the Euler equations to make the following integrals stationary. Change the independent variable, if needed, to make the Euler equation simpler.

${\mathbf{\int }}_{{\mathbf{t}}_{\mathbf{1}}}^{{\mathbf{t}}_{\mathbf{2}}}{\mathbf{s}}^{\mathbf{-}\mathbf{1}}\sqrt{{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{s}{\mathbf{\text{'}}}^{\mathbf{2}}}\mathbf{dt}\mathbf{,}\mathbf{}\mathbf{s}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{ds}}{\mathbf{dt}}$

Use the integral test to find whether the following series converge or diverge. Hint and warning: Do not use lower limits on your integrals.

10.$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{e}}^{\mathbf{n}}}{{\mathbf{e}}^{\mathbf{2}\mathbf{n}}\mathbf{+}\mathbf{9}}$

$\mathit{s}\mathit{i}\mathit{n}{\mathit{x}}^{\mathbf{2}}$

Use equation (1.8) to find the fraction that are equivalent to following repeating decimals.

10. 0.576923076923076923….

Given the measurements.

$\begin{array}{l}\mathbf{x}\mathbf{:}\mathbf{5}\mathbf{.8}\mathbf{,}\mathbf{6}\mathbf{.1}\mathbf{,}\mathbf{6}\mathbf{.4}\mathbf{,}\mathbf{5}\mathbf{.9}\mathbf{,}\mathbf{5}\mathbf{.7}\mathbf{,}\mathbf{6}\mathbf{.2}\mathbf{,}\mathbf{5}\mathbf{.9}\\ \mathbf{y}\mathbf{:}\mathbf{2}\mathbf{.7}\mathbf{,}\mathbf{3}\mathbf{.0}\mathbf{,}\mathbf{2}\mathbf{.9}\mathbf{,}\mathbf{3}\mathbf{.3}\mathbf{,}\mathbf{3}\mathbf{.1}\end{array}$

Find the mean value and the probable error of $\mathbf{2}\mathit{x}\mathbf{-}\mathit{y}\mathbf{,}{\mathit{y}}^{\mathbf{2}}\mathbf{-}\mathit{x}\mathbf{,}{\mathit{e}}^{\mathbf{y}}$and $\frac{\mathbf{x}}{{\mathbf{y}}^{\mathbf{2}}}$.

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\underset{\mathit{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\frac{\mathbf{1}\mathbf{-}{\mathbf{e}}^{{\mathbf{x}}^{\mathbf{3}}}}{{\mathbf{x}}^{\mathbf{3}}}$ .

The following alternating series are divergent (but you are not asked to prove this). Show that${\mathit{a}}_{\mathbf{n}}\mathbf{\to }\mathbf{0}$. Why doesn’t the alternating series test prove (incorrectly) that these series converge?

$$\begin{gathered} \mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{2}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{+}\frac{\mathbf{2}}{\mathbf{3}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{+}\frac{\mathbf{2}}{\mathbf{5}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{6}}\mathbf{+}\frac{\mathbf{2}}{\mathbf{7}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{8}}\mathbf{\cdots }\mathbf{} \\ \mathbf{\left(}\mathit{b}\mathbf{\right)}\frac{\mathbf{1}}{\sqrt{\mathbf{2}}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{+}\frac{\mathbf{1}}{\sqrt{\mathbf{3}}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{+}\frac{\mathbf{1}}{\sqrt{\mathbf{4}}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{+}\frac{\mathbf{1}}{\sqrt{\mathbf{5}}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{5}} \end{gathered}$$

In the partial differential equation

$\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{z}}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{2}}}\mathbf{-}\mathbf{5}\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{z}}{\mathbf{\partial }\mathbf{x}\mathbf{\partial }\mathbf{y}}\mathbf{+}\mathbf{6}\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{z}}{\mathbf{\partial }{\mathbf{y}}^{\mathbf{2}}}\mathbf{=}\mathbf{0}$

put $\mathit{s}\mathbf{=}\mathit{y}\mathbf{+}\mathbf{2}\mathit{x}\mathbf{,}\mathit{t}\mathbf{=}\mathit{y}\mathbf{+}\mathbf{3}\mathit{x}$ and show that the equation becomes role="math" localid="1665051128513" $\frac{{\mathbf{\partial }}^{\mathbf{2}}\mathbf{z}}{\mathbf{\partial }\mathbf{s}\mathbf{\partial }\mathbf{t}}\mathbf{=}\mathbf{0}$. Following the method of solving (11.6), solve the equation.