Is the Maclaurin series expansion of $\\sin x$ related to the inclusion-exclusion principle?

Question

When I see the alternating signs in the infinite series expansion of $\\sin x$, I'm reminded of the inclusion-exclusion principle. Could there be any way to visualize it in such a way? Also, is there an elementary reason why the Taylor series approximates a function? I've read the wikipedia entry but didn't understand it so it's greatly appreciated if someone can point me in the right direction.

Answer 1

Not exactly. It is more-so related to the Maclaurin Series for e^x. where x=-1. The same Maclaurin Series is useful in proving that the probability of obtaining a derangement is asymptotic. When I try to reason, without proof, why Taylor's Theorem works, I think about the linear approximation of a function (using the first derivative). If the linear approximation is good, then a quadratic approximation would be better, and a cubic approximation even better, so forth and so on. The reason the coefficients are as they are is because think about taking the integral of a power functions, you get xⁿ's integral is xⁿ⁺¹/(n+1); the integral of this is xⁿ⁺²/(x+1)(x+2). If you started from a linear term, you would have a denominator of 1*2*3***(n)*(n+1)=(n+1)!. A great visualization of this concept can be found here: https://www.youtube.com/watch?v=3d6DsjIBzJ4.

Answer 2

Tough question.

I don't know the inclusion-exclusion principle; therefore, I can't comment on that part of your question.

Of course, you need to understand a lot of calculus to understand the answer.

You need to look at the derivation of Taylor's theorem, especially the part dealing with the remainder.

As a start, look at the Mean Value theorem...because in some sense Taylor's theorem is an extension of the mean value theorem.

Now all that said, I need to refer you to a textbook for the complete explanation: "Calculus" by Michael Spivak.a pedagogic tour de force which I heartily recommend.