To control an agricultural pest called the medfly (Mediterranean fruit fly), $N$ sterilized male flies are released into the general fly population every day. If $s$ is the proportion of these sterilized flies that survive a given day, then $Ns^k$ will survive for $k$ days. If $s=$ 0.9 and 10,000 sterilized males are needed to control the medfly population in a given area, how many should be released every day?

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

$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n^3}$$

Use series to evaluate the limit.

$$\lim _{x \rightarrow 0} \frac{x-\ln (1+x)}{x^2}$$

The force due to gravity on an object with mass $m$ at a height $h$ above the surface of the earth is

$$F=\frac{m g R^2}{(R+h)^2}$$

where $R$ is the radius of the earth and $g$ is the acceleration due to gravity. Observe that if we approximate $F$ by the first term in the series, we get the expression $F \approx m g$ that is usually used when $h$ is much smaller than $R$. Use the Alternating Series Estimation Theorem to estimate the range of values of $h$ for which the approximation $F \approx m g$ is accurate to within one percent. (Use $R=6400$ km.)

The force due to gravity on an object with mass $m$ at a height $h$ above the surface of the earth is

$$F=\frac{m g R^2}{(R+h)^2}$$

where $R$ is the radius of the earth and $g$ is the acceleration due to gravity. Express $F$ as a series in powers of $h / R$.

Determine whether the sequence $a_n = \dfrac{2n - 3}{3n + 4}$ is increasing, decreasing, or not monotonic. Is the sequence bounded?

Use series to approximate the definite integral to within the indicated accuracy.

$$\int_0^{0.5} x^2 e^{-x^2} d x \quad(\mid \text{error} \mid<0.001)$$

Find a power series representation for the function and determine the interval of convergence.

$$f(x)=\dfrac{4}{2 x+3}$$

Find a power series representation for the following function and determine the interval of convergence.

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

Calculate the first ten terms of the sequence of partial sums. Does it appear that the series is convergent or divergent? Explain your reasoning.

$$\displaystyle\sum_{n=2}^{\infty} \frac{n}{\ln n}$$

Find the Taylor polynomial $T_3(x)$ for the function $f$ at the number $a$. Graph $f$ and $T_3$ on the same screen.

$$f(x)=\cos x, \quad a=\pi / 2$$

Find a formula for the general term $a_n$ of the sequence, assuming that the pattern of the first few terms continues.

$$\left\{-3,2,-\dfrac{4}{3}, \dfrac{8}{9},-\dfrac{16}{27}, \ldots\right\}$$

Find the Taylor polynomial $T_3(x)$ for the function $f$ centered at the number $a$. Graph $f$ and $T_3$ in the same viewing rectangle.

$$f(x)=e^{-x} \sin x, \quad a=0$$

A patient is prescribed a drug and is told to take one 100-mg pill every eight hours. After eight hours, about 5% of the drug remains in the body. What happens in the long run?