Test for convergence.

$$\sum_{j=1}^{\infty}(-1)^j \sin \left(\frac{\pi}{4 j}\right)$$

Define astigmatism.
How can it be corrected?

a. Suppose that $F_1=(5,0), F_2=(-5,0)$. and $P=(x, y)$. Write an equation that expresses the fact that $\left|P F_1-P F_2\right|=8$.

b. Simplify your equation to the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

For the given function below, graph $f$ and $f^{\prime}$. Then estimate the coordinates of the point where the tangent line to f is horizontal. If no such point exists, state that fact.

$$f(x)=\frac{x^3-1}{x^2+1}$$

Find all roots of the equation log z = iπ/2.

Evaluate $\frac{15 \pm 5 \sqrt{225}}{3}$.

(A) $-70$ and $80$

(B) $-20$ and $30$

(C) $20$ and $30$

(D) $70$ and $80$

Rewrite the sentence adding details to make it more interesting and descriptive.

We walked through the woods after it snowed.

¿Qué magnitud se representa por un vector que va de la posición inicial a la final indicando el cambio de posición de un cuerpo.

a) Desplazamiento.

b) Trayectoria.

c) Distancia.

d) Movimiento.

The following problem, show that the matrix A is nilpotent and then use this fact to find the matrix exponential $e^{\mathbf{A} t}$.

$$\mathbf{A}=\left[\begin{array}{ll} 1 & -1 \\ 1 & -1 \end{array}\right]$$

Express each rational number as a decimal.

$$\frac { 5 } { 7 }$$

Express the relation as a table, a graph , and a mapping. Then determine the domain and range. {(-2, 4), (-1, 3), (0, 2), (-1, 2)}

Factor completely. See the discussed examples.

$$6 x^2 y^2-2 x y^2-60 y^2$$

The standard form for $5^{10}$ is $9,765,625$. How can you use this fact to evaluate $5^{11}$ ?

Does there exist a power series $\sum _ { j = 0 } ^ { \infty } a _ { j } z ^ { j }$ that converges at z = 2+3i and divergesat z = 3 − i?