Best Explanation of Taylor Series on Internet!

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0:00A first-order approximation near p is $f(p)+f$′(p)($x-p$). It matches both the function value and slope at the center.

3:00Adding f″(p)($x-p$)²/2! matches the second derivative as well. Further coefficients use higher derivatives divided by the corresponding factorials.

6:00At $p=0$ this is the Maclaurin construction. For sine it yields $x-x$³/3!+x⁵/5!−⋯; in general, matching all derivatives alone does not guarantee that an infinite Taylor series equals the function away from its center.

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