It is quite exciting to think that these non-polynomial functions can be represented by infinite polynomial series. Examples include the functions \(e^x\), \(\sin(x)\) and \(\cos(x)\). In fact, the expansions for some functions are valid for all values of \(x\). What is truly remarkable is that for some functions the expansions constructed by the method outlined above are valid over a wider interval than \(\).
INFINITE POWERS SERIES
In the case of the series for \(\ln(1+x)\) it turns out that the infinite series converges to the correct value on the whole interval \(\), albeit rather slowly for some values of \(x\). It is possible to use tools of mathematical analysis to formally determine the interval of convergence for the infinite series, which will be different for each function \(f(x)\). By constructing the expansion to match the derivatives of \(f(x)\) at \(x=0\) we have only ensured that the representation is accurate in some local neighbourhood of this point. However, it is not guaranteed that the infinite sum \(p(x)\) will converge to \(f(x)\) on the whole interval \(\). In the above discussion we have assumed that \(|x|<1\) so that the powers of \(x\) grow successively smaller. We see that the linear approximation is an accurate representation of the curve in a very small domain close to \(x=0\), and that by adding the quadratic and cubic terms we successively improve the approximation on the domain \(\).
8.1 Comparison of asymptotic expansions to the curve \(y=e^x\) on the domain \(\). Therefore the leading terms are most significant in the expansion and the higher powers are “corrections”.īy way of example, consider the plots below, which show the curve \(y=e^x\) together with its linear, quadratic and cubic asymptotic expansions.įig. Notice that if \(x\ll 1\) (“much less than one”) the higher powers are successively smaller.
Import numpy as np import matplotlib.pyplot as plt x = np.
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