If $k$ is a positive integer, find the radius of convergence of the series

$$\sum_{n=0}^2 \frac{(n !)^k}{(k n) !} x^n$$

If

$$\mathbf { F } ( \mathbf { r } ) = \frac { c } { \| \mathbf { r } \| ^ { 3 } } \mathbf { r }$$

is an inverse-square field, and if

$$\sigma$$

is a closed orientable surface that surrounds the origin, then Gauss’s law states that the outward flux of F across

$$\sigma$$

is ___. On the other hand, if

$$\sigma$$

does not surround the origin, then it follows from the Divergence Theorem that the outward flux of F across

$$\sigma$$

is ___.

Radial fields Consider the radial vector field $\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^p}=\frac{\langle x, y, z\rangle}{\left(x^2+y^2+z^2\right)^{p / 2}}$. Let $S$ be the sphere of radius $a$ centered at the origin.

Use a surface integral to show that the outward flux of $\mathbf{F}$ across $S$ is $4 \pi a^{3-p}$. Recall that the unit normal to the sphere is $\mathbf{r} /|\mathbf{r}|$.

Use the Divergence Theorem to calculate the surface integral ∫∫sF·dS; that is, calculate the flux of F across S. F=|r|^2r, where r = xi + yj + zk, S is the sphere with radius R and center the origin.

Find the slope of the curve at each indicated point. r = -1 + sin θ, θ = 0,π

Find the radius of convergence and interval of convergence of the series.

$$\sum_{n=1}^n \frac{n x^n}{1 \cdot 3 \cdot 5 \cdots \cdots(2 n-1)}$$

Determine whether the sequence converges or diverges. If it converges, find the limit.

$$\left\{\frac{3+(-1)^n}{n^2}\right\}$$

Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?

Determine whether the series is convergent or divergent.

$$\sum_{n-1}^x \frac{1+2^n}{1+3^n}$$

Approximate the sum of the series to the indicated accuracy.

$\sum_{n-1}^{\infty} \frac{(-1)^{n-1}}{n^6}$ (five decimal places)

Determine whether the sequence converges or diverges. If it converges, find the limit.

$$\left\{\frac{\pi^n}{3^n}\right\}$$

Determine whether the series is convergent or divergent. If it is convergent, find its sum.

$$\sum_{n=1}^x\left[2(0.1)^n+(0.2)^n\right]$$

Does $\sum (-1)^n \ln \left(2n^{1/n}\right)$ converge or diverge?

Consider the sequence defined recursively by $a_1=1$ and, for $n>1, a_n=\sum_{i=1}^{n-1} a_i$. Write down the first six terms of this sequence, guess a formula for $a_n$ valid for $n \geq 2$, and prove your answer.