anonymous
  • anonymous
help!!
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
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 ?
anonymous
  • anonymous
like \(f_1(x)= log (1+sin x )\) \(f_2(x)=f_1+2 log (1+sin x/2 )\) ... ?!
klimenkov
  • klimenkov
Use this: \(\sin x = x + O(x^3), \, x\rightarrow 0\). \[\lim_{n\rightarrow \infty}n\log\left(1 + \sin\left( x/n\right)\right) = \lim_{n\rightarrow \infty}\log\left(1 + x/n\right)^n\]

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anonymous
  • anonymous
it would be something like this :- |dw:1401617819184:dw|
anonymous
  • anonymous
@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 ??
klimenkov
  • klimenkov
\(x\) is a fixed point and \(n\rightarrow \infty\). So it will converge for every finite \(x\). \[\log e^x = x.\]
anonymous
  • anonymous
@klimenkov thank you..
anonymous
  • anonymous
@BSwan thank you

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