Okay, I have this function defined as an infinite series:
f(x) = \sum_{n=1}^{\infty}\frac{\sin(nx)}{x+n^4}
which is converges uniformly and absolutely for x > 0. I have shown that f is continous and has a derivative for x > 0. Now I have to show that f(x) \rightarrow 0 as x \rightarrow...
Now, I've done the b-question in a rigorous and satisfying way. Thanks for helping.
That is actually the definition in my book (pretty non-intuitive, grrr...). The radius of convergence is just the radius of the circle in the complex plane inside which the power series converges - I don't...
I can of course throw the i outside the summation and conclude that the series diverges? Is that a valid step when considering infinite sums? Remember, it has to be rigorous (this is a course, where we started out proving that 1+1=2...)
Hehe, my book defines radius of convergence as...
For z = i I get:
\sum_{n=0}^{\infty}\frac{i}{2n+1}.
Most of the series theorems in my book involves only series with real, not complex, terms. The only theorem I can think of is the one that states that for a series to be convergent, the individual terms must tend to 0 - they do in this...
I really need help with this exercise. Consider the power series
\sum_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{2n+1}.
for z\in\mathbb{C}.
I need to answer the following questions:
a) Is the series convergent for z = 1?
This is easy; just plug in z = 1 and observe that the alternating...
We want to show that \sqrt{n^2+2n} - (n+1) \rightarrow 0, so we rewrite the expression:
\sqrt{n^2+2n} - (n+1) = \frac{\left(\sqrt{n^2+2n} - (n+1)\right)\left(\sqrt{n^2+2n} + (n+1)\right)}{\sqrt{n^2+2n} + (n+1)} = \frac{-1}{\sqrt{n^2+2n} + (n+1)}
It is obvious that the final expression...
Hi,
I have to find the expectation values of xp and px for nth energy eigenstate in the 1-d harmonic oscillator. If I know <xp> I can immediately find <px>since [x,p]=ih. I use the ladder operators a_{\pm}=\tfrac1{\sqrt{2\hslash m\omega}}(\mp ip+m\omega x) to find <xp>, but I get a complex...
Yep, converges towards -ln(2). It's obvious how the hint rules out absolute convergence, since \sum_{n=1}^{\infty}\frac1{n} is divergent. But the alternating one?
I am given this series:
\sum_{n=1}^\infty\frac{2n}{n^2+1}z^n.
First I have to find the radius of convergence; I find R = 1. Then I have to show that the series is convergent, but not absolutely convergent, for z = -1, i.e. that the series
\sum_{n=1}^\infty(-1)^n\frac{2n}{n^2+1}
is...
Hi,
I have to show that the function
f(x) = \sum_{n=1}^{\infty}\frac1{x^2+n^2}
tends to 0 as x \rightarrow \infty, i.e. \lim_{x\rightarrow\infty}f(x) = 0. How can I do this?
There is a hint that says I should use the inequality f(x) \leq \sum_{n=1}^N\tfrac1{x^2+n^2} +...
I am having trouble with an exercise from Griffiths "Introduction to Quantum Mechanics". The exercise is this:
"Suppose you add a constant V_0 to the potential energy. In classical mechanics this doesn't change anything, but how about quantum mechanics? Show that the wave function picks up a...
nope.. I can't get the right answer.. I'd rather not post my working, since it's is very messy :/ I'm not asking someone to do the calculations; I would just like a general (the easiest) way to deal with such problems...
Hi!
I have to calculate the Fourier coefficients c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx and the Fourier series for the following function:
f(x)=
\begin{cases}
\frac{2}{\pi}x + 2 & \text{for } x\in \left[-\pi,-\pi/2\right]\\
-\frac{2}{\pi}x & \text{for } x\in...
Thanks for answering.
I'm not sure; do you mean I should try to fit the data to functions like this:
f_n(t) = A_1e^{\lambda_1 t} + A_2e^{\lambda_1 t} + \ldots + A_ne^{\lambda_n t}
for various n (corresponding to the number of different isotopes) and see which fit is the best. If so...