Show that the series $\sum_{j=1}^{\infty}\left(1-2^{-j}\right) / j$ diverges.

Show that $\oint_{C} \frac{-y d x+x d y}{x^{2}+y^{2}}=2 \pi$, where C is the circle $x^{2}+y^{2}=a^{2}$.

What does the equation $(x-h)^{2}+(y-k)^{2}+(z-j)^{2}=r^{2}$ represent? What do $h, k, j,$ and $r$ represent?

Use Newton's method to find all roots of the equation $2 \cos x=2-x$ correctly to six decimal places.

The midpoint M and one endpoint of $\overline{J K}$ are given. Find the coordinates of the other endpoint. $J(2,16)$ and $M\left(-\frac{9}{2}, 7\right)$

$$\begin{array}{l}\mathrm{m} \angle H J L=116^{\circ} \\ \text { Find } \mathrm{m} \angle H J K\end{array}$$

existing "in fact" rather than officially or legally

Find $\mathbf{a}\cdot \mathbf{b}$, where $\mathbf{a} = 4\mathbf{i }+ 8\mathbf{j}$ and $\mathbf{b} = -2\mathbf{i }+ 3\mathbf{j}$.

Solve for $x$.

$$\frac{3 x+2}{x^2+x-2}=\frac{4}{x+2}-\frac{2}{1-x}$$

Generate Bode magnitude and phase plots (straight-line approximations) for the following voltage transfer functions. a) ${H}(\omega)=\dfrac{j 100 \omega}{10+j \omega}$

b) ${H}(\omega)=\dfrac{0.4(50+j \omega)^2}{(j \omega)^2}$

c) ${H}(\omega)=\dfrac{(40+j 80 \omega)}{(10+j 50 \omega)}$

d) ${H}(\omega)=\dfrac{(20+j 5 \omega)(20+j \omega)}{j \omega}$

If $\mathbf{a}=2 \mathbf{i}+5 \mathbf{j}$ and $\mathbf{b}=4 \mathbf{i}-\mathbf{j}$, find the vector or number corresponding to

$$\|\mathbf{a}\|-\|\mathbf{b}\|$$

Do you find any irony in the fact that Ezekiel Cheever is the one who arrests Elizabeth Proctor? Why or why not?

Gianna plotted these points and then connected the points in order from J to N and then back to J to show the shape of her room. Draw the room on the coordinate plane. What is the area of Gianna's room? J(1, 0) K(1, 6) L(9, 6) N{6, 3).

Identify the initial amount a and the rate of growth r (as a percent) of the exponential function. Evaluate the function when t = 4. Round your answer to the nearest tenth. $f(t)=4.2(1.9)^{t}$