Chapter 1: Infinite Series, Power Series

Find out whether infinity is a regular point, an essential singularity, or a pole (and if a pole, of what order) for each of the following functions. Find the residue of each function at infinity,$\mathbf{tan}\frac{\mathbf{1}}{\mathbf{z}}$

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

$\mathbf{\sum }_{\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{1}}{\mathbf{n}{\left(1+ lnn\right)}^{\frac{\mathbf{3}}{\mathbf{2}}}}$

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.

$\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\frac{{\mathbf{sin}}^{\mathbf{2}}\mathbf{2}\mathbf{x}}{{\mathbf{x}}^{\mathbf{2}}}$ .

$\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\sqrt{\mathbf{x}}\mathbf{\right)}}{\sqrt{\mathbf{x}}}\mathbf{,}\mathit{x}\mathbf{>}\mathbf{0}$

Find$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}$ explicitly if $\mathit{y}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{1}}\frac{{\mathbf{e}}^{\mathbf{x}\mathbf{u}}\mathbf{-}\mathbf{1}}{\mathbf{u}}\mathit{d}\mathit{u}$.

By the method used to obtain (12.5)[which is the series(13.1)below], verify each of the other series (13.2)to (13.5)below.

If $\mathit{y}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{x}}\mathit{s}\mathit{i}\mathit{n}{\mathit{t}}^{\mathbf{2}}\mathbf{d}\mathit{t}$, find $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}$.

Find the interval of convergence, including end-point tests$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{x}}^{\mathbf{n}}{\mathbf{n}}^{\mathbf{2}}}{{\mathbf{5}}^{\mathbf{n}}\left(n^2+1\right)}$

The vectors $\mathbf{A}=a\mathbf{i}+b\mathbf{j}$and $\mathbf{B}=c\mathbf{i}+d\mathbf{j}$form two sides of a parallelogram. Show that the area of the parallelogram is given by the absolute value of the following determinant.

role="math" localid="1664258424007" $\left|\begin{array}{cc}\text{a}& \text{b}\\ \text{c}& \text{d}\end{array}\right|$

In a water purification process, one-nth of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage. Show that if$\mathit{n}\mathbf{=}\mathbf{2}$, the water can be made as pure as you like, but that$\mathit{n}\mathbf{=}\mathbf{2}$ if, at least one-half of the impurity will remain no matter how many stages are used.