Well, I finally got the answer and it was very simple in fact!
It's only necessary to expand all the terms enclosed in the parenthesis and truncate the expansion in 5th order.
Homework Statement
Exercise 5.6.24 from Arfken's Essential Mathematical Methods for Physicists. You have a function tabulated at equally spaced values of the argument:
\left\{ \begin{array}{c}
y_n=y(x_n)\\x_n = x+nh \end{array}\right.
Show that the linear combination
\frac{1}{12h}\left\{...
Thanks, lurflurf. Your idea was of a great help. But the uniform convergence is still giving me headache. I was thinking that if I could prove that the sum \sum \frac{1}{1+x^n} is a monotonically decreasing function, for any value of n, then the bigger the x, smaller the sum, so It would be...
Homework Statement
For what range of positive values of x is \sum_{n=0}^\infty \frac{1}{1+x^n}
(a) convergent
(b) uniformly convergent
Homework Equations
The Attempt at a Solution
I didn't figure out how to separate convergence and uniformly convergence for this series.
My idea...
Thank you very much, statdad and lurflurf. I think that I finally understood what the author wanted to say. The product of series presents us to a dramatically new form of seeing the "tails" of the series. By the way, interesting find this google's books. Sorry for my late reply.
While reading the section about algebra of series of Arfken's essential Mathematical Methods for Physicist, I faced an intriguing demonstration of the product convergence theorem concerning an absolutely convergent series \sum u_n = U and a convergent \sum v_n = V . The autor assured that if...
Homework Statement
Test the convergence of the series for the surface charge density:
\sum^{\infty}_{s=0}(-1)^s(4s+3)\frac{(2 s -1)!}{(2s)!}
Homework Equations
(2s-1)! = \frac{(2s)!}{2^s s!};
(2s)! = 2^s s!
Stirling's asymptotic formula for the factorials:
s! = \sqrt{2 \pi s}s^s...
Homework Statement
The function f(z) is analytic. Show that the derivative of f(z) with respect to z* does not exist unless f(z) is a constant.
Hint: Use chain rule and take x = (z+z*)/2, y = (z-z*)/2.
Homework Equations
\frac{d f}{d z*} = \frac{\partial f}{\partial x}\frac{\partial...
No, it didn't change; I understand what you mean. Ok I will think about how establish a relation between the old and the new eccentricity using the fact that the mass of the planet is constant; I truly believe that the solution resides in considering this.
Homework Statement
A planet is in a circular orbit about a star that explodes, shedding 2% of its mass in an expanding spherical shell. Find the eccentricity of the new orbit of the planet, which otherwise is not affected by the shell.
Homework Equations
\sqrt{1-\varepsilon^2} =...
Considering that this plan is EXACTLY above the North pole, and that in the initial instant of time the jet plane is flying through this radius vector, the answer would depend of a time interval. What I could do?
I mean, the plane would be a distance dx = v dt \hat{\mathbf{x}} in...
Homework Statement
A jet plane flies due south over the north pole with a constant speed of 500 mph. Determine the angle between a plumb line hanging freely in the plane and the radius vector from the center of the Earth to the plane above the north pole.
Hint, assume that the Earth's angular...
I'm trying to solve exercise number 2 of Cohen's Quantum Mechanics, vol1.
The item (b) demands the normalization of the functions but I can't do it because I need one more relation between the constants A1 and A1'. Does anyone have any idea?