Testwiki:Reference desk/Archives/Mathematics/2014 August 22

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August 22

Taylor /Maclaurin Series number of roots

When applying Taylor Series or any other expansions to a [[|Function (mathematics)|function]] such as $\frac{1}{x}$, then we will have a polynomial with infinite number of terms. Can we then consider this infinite terms polynomial to have a degree of n the approaches infinity and so, has infinite number of roots? Or is considered as a mathematical fallacy since we applied the series to a point that is not necessarily near the roots?--Almuhammedi (talk) 06:11, 22 August 2014 (UTC)

The problem is that you're playing with infinity here. The fundamental theorem of algebra strictly applies only to finite degree polynomials. It also says nothing about whether the set of roots of any of the partial sums must have a limit as n→infinity. Naturally there are cases such as the trigonometric functions that do have infinitely many roots. But this is certainly not true in general, such as with the exponential function, which is nowhere zero.--Jasper Deng (talk) 06:59, 22 August 2014 (UTC)

Here's an interesting paper on the location of the n roots of the n^th partial sum of the Taylor expansion of the exponential function as n goes to infinity: Ian Zemke Zeroes of the Partial Sums of the Exponential Function. --catslash (talk) 12:40, 22 August 2014 (UTC)