Your name: Rachelle Howell

Homework 1

Due in class on Thursday, September 9

1. The great Russian mathematician Andrey Kolmogorov made his first mathematical discovery when he was 5 years old. He noticed that

$1=1^2$

$1 + 3 = 4 = 2^2$

$1 + 3 + 5 = 9 = 3^2$

$1 + 3 + 5 + 7 = 16 = 4^2$

...

$\displaystyle \sum\limits_{k=1}^n (2\,k-1) = n^2$

Using the method of mathematical induction, prove that the observation of the little Kolmogorov is true for every $n \ge 1$ .

\rule{6cm}{1mm}

figure = sum([polygon([[0,0],[0,k],[k,k],[k,0]],hue=k/10) for k in range(10,0,-1)])
figure.show(aspect_ratio=1)

2. Consider the infinite series

$S = \displaystyle \frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \cdots + \frac{n}{(n+1)!} + \cdots$

(a) Using one of the convergence tests, show that the series converges.

(b) Prove that the partial sum is

$\displaystyle S_n = \sum\limits_{k=1}^{n} \frac{k}{(k+1)!} = 1 - \frac{1}{(n+1)!}$

(c) Find the value of $S$ .

3. Prove the formula for the sum of a geometrical progression

$\displaystyle 1 + x + x^2 + \cdots + x^{n-1} = \frac{1-x^n}{1-x}$

What is the radius of convergence for the infinite series $\displaystyle \sum\limits_{k=1}^{\infty} x^{k-1}$ ?

k,n=var('k,n')
sum(x^(k-1),k,1,n)

(x^n - 1)/(x - 1)

4. Use an appropriate convergence test to check if the following series are converging or diverging.

(a) $\displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n\,2^n}$

(b) $\displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{\ln (n+1)}$

(c) $\displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{\sqrt{n^2+4}}$

(d) $\displaystyle \sum\limits_{n=1}^{\infty} \frac{n!}{(2\,n+1)!}$

(e) $\displaystyle \sum\limits_{n=1}^{\infty} \frac{(-3)^n}{n^2}$

5. Seismic reflection traveltime as a function of the source-receiver offset on the surface $t(x)$ can be expressed as an infinite series

$t(x) = a_0 + a_1\,x^2 + a_2\,x^4 + \cdots$

The series has only even powers of $x$ , because $t(x)$ is an even function (the traveltime remains the same when the source and receiver get exchanged and the offset becomes negative.) Different functional approximations have been proposed to describe the behavior of the traveltime function avay from the zero offset.

(a) Consider an approximation

$\displaystyle t(x) \approx \left(1-\frac{1}{s}\right)\,t_0 + \frac{1}{s}\,\sqrt{t_0^2 + s\,\frac{x^2}{v^2}}$

By expanding it in a power series around $x=0$ , express the coefficients $t_0$ , $v$ , and $s$ in terms of $a_0$ , $a_1$ , and $a_2$ .

(b) Consider an alternative approximation

$\displaystyle t(x) \approx \sqrt{t_0^2 + \frac{x^2}{v^2} -
\frac{q\,x^4}{\displaystyle v^4\,\left[t_0^2 + (1+q)\,\frac{x^2}{v^2}\right]}}$

By expanding it in a power series around $x=0$ , express the coefficients $t_0$ , $v$ , and $q$ in terms of $a_0$ , $a_1$ , and $a_2$ .

(c) Can you suggest yet another approximation with similar properties?

t0,v=var('t0,v')
taylor(sqrt(t0^2+x^2/v^2),x,0,4)

t0 + 1/2*x^2/(t0*v^2) - 1/8*x^4/(t0^3*v^4)

p=var('p')
eta2=0.2
t_exact=parametric_plot((p/((1-eta2*p^2)^2*sqrt(1-p^2/(1-eta2*p^2))),
 ((1-eta2*p^2)^2+eta2*p^4)/((1-eta2*p^2)^2*sqrt(1-p^2/(1-eta2*p^2)))),
 (p,-0.8,0.8),thickness=2)
t_approx=plot(sqrt(1+x^2),(x,-2,2),color='green')
t=t_exact+t_approx
t.show(xmin=-2,xmax=2)

References

Alkhalifah, T., and I. Tsvankin, 1995, Velocity analysis for transversely isotropic media: Geophysics, 60, 1550-1566.

Castle, R. J., 1994, Theory of normal moveout: Geophysics, 59, 983-999.

Fomel, S., and A. Stovas, 2010, Generalized nonhyperbolic moveout approximation: Geophysics, 75, U9–U18.