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oksuz_
help!!
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 ?
like \(f_1(x)= log (1+sin x )\) \(f_2(x)=f_1+2 log (1+sin x/2 )\) ... ?!
Use this: \(\sin x = x + O(x^3), \, x\rightarrow 0\). \[\lim_{n\rightarrow \infty}n\log\left(1 + \sin\left( x/n\right)\right) = \lim_{n\rightarrow \infty}\log\left(1 + x/n\right)^n\]
it would be something like this :- |dw:1401617819184:dw|
@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 ??
\(x\) is a fixed point and \(n\rightarrow \infty\). So it will converge for every finite \(x\). \[\log e^x = x.\]