How are we helped to allocate goods and services by prices?

If the sequence is convergent, find its limit. If it is divergent, explain why.

$a_n=\frac{n^2}{n+1}$

Is the following series convergent or divergent? 1 + 1/2·19/7 +2!/3²(19/7)² + 3!/4³(19, 7)³ + 4!/5^4(19, 7)^4 + ...

By using Laplace transforms, solve the following differential equations subject to the given initial conditions

$$y^{^{\prime \prime }}+y=5\sinh \left( 2t\right) ,~~~~~y_{0}=0,~y_{0}^{^{\prime }}=2.$$

An uncharged capacitor is charged by a battery through a resistor (call this charging situation #1). The capacitor is then completely discharged and recharged (using a different battery but the same resistor) to twice the final charge as in situation #1 (call this charging situation #2). How do the charging time constants compare: $\text { (a) } \tau_{1}=2 \tau_{2},(\mathrm{b}) \tau_{1}=\tau_{2},(\mathrm{c}) \tau_{1}=\frac{1}{2} \tau_{2} ?$

Show that only half the total energy drawn from a battery in charging an $R C$ circuit ends up stored in the capacitor. (Hint: What happens to the rest? You'll need to integrate.)

Give an example of a region in the plane which gives the same volume whether rotated about the x-axis or the y-axis.

Find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum.

$$\sum_{n=0}^{\infty}(-1)^n \frac{1}{3^{2 n+1}(2 n+1) !}$$

Horseshoe crabs have changed little in structure from the time they first showed up in the fossil record. Which pattern of evolution do horseshoe crabs likely follow—gradualism or punctuated equilibirum? Explain your answer.

Determine whether the given series is convergent or divergent.

$$\sum^{\infty}_{n=1}\frac{1}{n\sqrt{n}}$$

Determine whether the series converges absolutely, converges conditionally, or diverges. Justify each answer. State the value of any convergent series for which it is possible.

$$\sum_{n=1}^{\infty} \frac{n !}{n^2}$$

Find the Maclaurin series for the function given. Assume that a is any constant, and that all the derivatives of all orders exist at x = 0.

$$\sin ^{2} x$$

The battery of figure is rated at $81\ \mathrm{Ah}$ at $5^{\circ} \mathrm{C}$. What is the expected life (in hours) at a current draw of $5 \mathrm{~A}$ at $-15^{\circ} \mathrm{C}$ ?

If the sequence is convergent, find its limit. If it is divergent, explain why.

$a_n=\frac{n}{5 n+1}$