Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

oksuz_

  • one year ago

help!!

  • This Question is Closed
  1. oksuz_
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    show that the sequence of function given by \[fn(x)=nlog(1+\sin(x/n)) \] converges point wise to the identity function f(x)=x anyone has an idea ?

  2. BSwan
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    like \(f_1(x)= log (1+sin x )\) \(f_2(x)=f_1+2 log (1+sin x/2 )\) ... ?!

  3. klimenkov
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Use this: \(\sin x = x + O(x^3), \, x\rightarrow 0\). \[\lim_{n\rightarrow \infty}n\log\left(1 + \sin\left( x/n\right)\right) = \lim_{n\rightarrow \infty}\log\left(1 + x/n\right)^n\]

  4. BSwan
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    it would be something like this :- |dw:1401617819184:dw|

  5. oksuz_
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @klimenkov at your solution last part equals e^x right? then when x goes to infinite e^x also goes to infinite.. should we consider identity function's interval here that is convergent ??

  6. klimenkov
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    \(x\) is a fixed point and \(n\rightarrow \infty\). So it will converge for every finite \(x\). \[\log e^x = x.\]

  7. oksuz_
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @klimenkov thank you..

  8. oksuz_
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @BSwan thank you

  9. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy