# 3.5: Problems

• • Contributed by Marcia Levitus

Problem $$\PageIndex{1}$$

Expand the following functions around the value of $$x$$ indicated in each case.

In each case, write down at least four terms of the series, and write down the result as an infinite sum.

• $$\sin{(ax)}$$, $$x=0$$, $$a$$ is a constant
• $$\cos{(ax)}$$, $$x=0$$, $$a$$ is a constant
• $$e^{ax}$$, $$x=0$$, $$a$$ is a real constant
• $$e^{-ax}$$, $$x=0$$, $$a$$ is a real constant
• $$\ln{(ax)}$$, $$x=1$$, $$a$$ is a real constant

Problem $$\PageIndex{2}$$

Use the results of the previous problem to prove Euler’s relationship:

$e^{ix}=\cos x + i \sin x \nonumber$

Problem $$\PageIndex{3}$$

The osmotic pressure ($$\pi$$) of a solution is given by

$-RT \ln x_A=\pi V_m \nonumber$

where $$V_m$$ is the molar volume of the pure solvent, and $$x_a$$ is the mole fraction of the solvent.

Show that in the case of a dilute solution

$RT x_B \approx \pi V_m \nonumber$

where $$x_B$$ is the mole fraction of the solute. Remember that the mole fractions of the solute and the solvent need to add up to 1.

Note: you may use any of the results you obtained in Problem $$\PageIndex{1}$$.

Problem $$\PageIndex{4}$$

The following expression is known as the Butler-Volmer equation, and it is used in electrochemistry to describe the kinetics of an electrochemical reaction controlled solely by the rate of the electrochemical charge transfer process.

$j=j_0({e^{(1-\alpha)f\eta}-e^{-\alpha f \eta}}), ~ 0<\alpha<1 \text{ and } f>0, \eta>0 \nonumber$

Show that $$j \approx j_0 f \eta$$ when $$f \eta <<1$$.

Note: you may use any of the results you obtained in Problem $$\PageIndex{1}$$.

Problem $$\PageIndex{5}$$

The energy density of black-body radiation ($$\rho$$) at temperature T is given by Plank’s formula:

$\rho(\lambda)=\frac{8\pi h c}{\lambda^5}[e^{hc/\lambda k T}-1]^{-1} \nonumber$

where $$\lambda$$ is the wavelength, $$h$$ is Plank’s constant, and $$c$$ is the speed of light. Show that the formula reduces to the classical Rayleigh-Jeans law $$\rho = 8\pi kT/\lambda^4$$ for long wavelengths ($$\lambda \rightarrow \infty$$).

Hint: Define a variable $$\nu = \lambda^{-1}$$ and solve the problem for $$\nu \rightarrow 0$$.

Note: you may use any of the results you obtained in Problem $$\PageIndex{1}$$.

Problem $$\PageIndex{6}$$

Use series to prove that $$\sum \limits _{k=0} ^\infty{\frac{\lambda^k e^{-\lambda}}{k!}}=1$$, $$\lambda$$ is a positive real constant.

Problem $$\PageIndex{7}$$

Write down the equation of a straight line that provides a good approximation of the function $$e^x$$ at values close to $$x = 2$$.

Problem $$\PageIndex{8}$$

Use a Taylor expansion around $$a$$ to prove that $$\ln{x} = \ln{a}+\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n a^n}(x-a)^n$$