Chapter 1: Infinite Series, Power Series

Use either Stokes' theorem or the divergence theorem to evaluate each of the following integrals in the easiest possible way.

$\mathbf{\iint }\mathbf{curl}\mathbf{(}\mathit{y}\mathbf{i}\mathbf{+}\mathbf{2}\mathbf{j}\mathbf{)}\mathbf{·}\mathbf{n}\mathit{d}\mathit{\sigma }$,where $\mathit{\sigma }$is the surface in the first octant made up of part of the plane $\mathbf{2}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{3}\mathit{y}\mathbf{}\mathbf{+}\mathbf{}\mathbf{4}\mathit{z}\mathbf{}\mathbf{=}\mathbf{}\mathbf{12}$, and triangles in the (x,z) and (y,z) planes, as indicated in the figure.

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{x}}^{\mathbf{2}\mathbf{n}}}{{\mathbf{2}}^{\mathbf{n}}{\mathbf{n}}^{\mathbf{2}}}$

Use equation (1.8) to find the fraction that are equivalent to following repeating decimals. 4. 0.818181….

First simplify each of the following numbers to the $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

${\mathit{i}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{i}\mathbf{+}\mathbf{1}$

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).${\mathbf{e}}^{\mathbf{sinx}}\mathbf{-}\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{3}}}\mathbf{In}\mathbf{(}\mathbf{1}\mathbf{+}{\mathbf{x}}^{\mathbf{3}}{\mathbf{e}}^{\mathbf{x}}\mathbf{)}\mathbf{at}\mathbf{}\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{00035}$.

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find $\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{r}\mathbf{,}\mathit{\theta }$in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

$\mathbf{-}\sqrt{\mathbf{3}}\mathbf{+}\mathit{i}$

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Solve the algebraic equation

$D^2+(1+2i)D+i-1=0$

(Note the complex coefficients) and observe that the roots are complex but not complex conjugates. Show that the method of solution of (5.6) (case of unequal roots) is correct here, and so find the general solution of

$y^{\text{'}\text{'}}+(1+2i)y^{\text{'}}+(i-1)y=0.$

Solve $y^{\text{'}\text{'}}+(1-i)y^{\text{'}}-iy=0$. Hint: See Chapter 2, Section 10, for a method of finding the square root of a complex number.

Show that each of the following matrices is orthogonal and find the rotation and/or reflection it produces as an operator acting on vectors. If a rotation, find the axis and angle; if a reflection, find the reflecting plane and the rotation, if any, about the normal to that plane.

$\frac{\mathbf{1}}{\mathbf{2}}\left(\begin{array}{ccc}-1& -1& \sqrt{2}\\ 1& 1& \sqrt{2}\\ \sqrt{2}& -\sqrt{2}& 0\end{array}\right)$

Test for convergence:$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{2}}^{\mathbf{\infty }}\frac{{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}}{\mathbf{1}\mathbf{+}{\mathbf{n}}^{\mathbf{2}}}$