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let be a function given by $ f^{-1} (x) = \sqrt x + G(x) $

with $ G(x)= \sum_{n=0}^{N}a_{n} \sin(nx+ \pi/4) $ finite fourier series with N big

my questio is , if for $ x \rightarrow \infty$ the function can be asymptotically defined by $ f(x) \sim x^{2} $

on condition that for every 'x' $ G(x) \le \le \sqrt x$ so the most importan term is the SMOOTH term defined by the square root of 'x'

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Well, $G(x)$ is obviously bounded (for example by $\sum_{n=0}^N |a_n|$, there might be a tigther one), so for large $x$, the relative error will go to zero. Not the absolute error, though.

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yo mean $ e_{relative}= e_{real}-e_{error}/e_{real} $ and $ e_{absolute}= e_{real}-e_{error} $ ?? thanks for your answer fgp –  Jose Garcia Oct 9 '12 at 20:26
    
@JoseGarcia Yes. –  fgp Oct 9 '12 at 20:52
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