Search results

  1. B

    Series again

    No, I can't find any relevant theorems to apply. I'm lost at sea.
  2. B

    Series again

    Maybe I'm a little slow here, but \sum_{n=1}^{\infty}\frac1{n^4} does not equal 0, so how come?
  3. B

    Series again

    Hmm.. How does that show that f(x) -> 0 as x -> infinity?
  4. B

    Series again

    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...
  5. B

    Power series

    Yes, since it converges at z=1 and diverges at z=i, R must be 1! :) You're a genius, thanks!
  6. B

    Power series

    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...
  7. B

    Power series

    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...
  8. B

    Power series

    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...
  9. B

    Power series

    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...
  10. B

    Limits using basic analysis theorems and logic?

    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...
  11. B

    Limits using basic analysis theorems and logic?

    Thanks guys. Hurkyl was right; there actually was an example dealing with this in the back of the book. I can't believe I missed it.
  12. B

    Limits using basic analysis theorems and logic?

    Hi, I need help again. How can I show that \sqrt{n^2+2n}-n \rightarrow 1 for n\rightarrow\infty using basic analysis theorems and logic? Any ideas?
  13. B

    Expectation value, harmonic oscillator

    Well, xp is not hermitian, I see your point, dexter. My expression for xp is i\hslash/2({a_+}^2 + {a_-}^2 + 1).
  14. B

    Expectation value, harmonic oscillator

    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...
  15. B

    Math (series)

    It diverges. It's the alternating series I'm having trouble with.
  16. B

    Math (series)

    I'm sorry, I'm an idiot. The hint says \frac{2n}{n^2+1}\geq\frac1{n}, that is what's baffling me. Please bear with me :yuck:
  17. B

    Math (series)

    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?
  18. B

    Math (series)

    No. I guess we're not supposed to use that?
  19. B

    Math (series)

    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...
  20. B

    Math problem (analysis)

    I can let N -> infinity and thereby make the second term vanish but then what about the first term which also depends on N?
  21. B

    Math problem (analysis)

    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} +...
  22. B

    Quantum Mechanics problem

    I found out. Pretty simple... Thanks anyway, I'll be needing your help in the future... I'll sketch my ideas in the future, I'm pretty lazy. Sorry :/
  23. B

    Quantum Mechanics problem

    That's the question :)
  24. B

    Quantum Mechanics problem

    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...
  25. B

    Fourier series

    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...
  26. B

    Fourier series

    no.. f(pi/2) = -f(pi/2) => f is odd?
  27. B

    Fourier series

    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...
  28. B

    Difficult nuclear physics exercise

    Weee! I managed to show that n = 2 using Matlabs curve fitting toolbox. Thank you very much for helping me.
  29. B

    Difficult nuclear physics exercise

    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...
Top