Calculus 2, Lec 26B: The Most MAGICAL 🧙‍♂️ & FUN 🎉 Taylor Series Examples

In Calculus, it's like magic when we find the Taylor series (a.k.a. Maclaurin series) for (1+x)^(-2) in 3 ways: 1) by the binomial series formula, 2) by multiplication of geometric series, and 3) by differentiation of a geometric series. Then we enjoy finding the series for ln(1+x) is by integrating a geometric series. Then the series for e^x, sin x, and cos x are found by differentiation. This is all good to know for AP Calculus BC. These are all fundamental examples.

#calculus2 #apcalculusbc #taylorseries

Links and resources
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(0:00) Taylor series for f(x)=(1+x)^(-2) by Binomial Series Formula
(1:46) Taylor series for f(x)=(1+x)^(-2) by multiplication of series term-by-term
(5:04) History (Crisis in Mathematical Analysis)
(7:25) Taylor series for f(x)=(1+x)^(-2) by differentiation of series term-by-term
(9:58) Taylor series for f(x)=ln(1+x) by integration of series term-by-term
(12:17) Taylor series for f(x)=e^x by Taylor series formula
(14:12) Ratio Test to confirm convergence for all x (infinite radius of convergence)
(17:24) Taylor series for f(x)=sin(x) by Taylor series formula
(22:00) Taylor series for f(x)=cos(x) by differentiation of series term-by-term

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[jesseaustin]

I've not used Taylor Series since my high school AP Calc test in 07/08 year (AB 5, BC 3. Didn't think 1minus x or x minus 1 diff denominator wouldnt be given especi free response).
I need advance maths now, after not being able to afford college and developing schizophrenia, which is less annoying now; medicated.

ChatGPT just said itd be amazing basically, if its correct, and I want to use this science, which gpt said tensors and field functions are what I'd need.. I want to mess around with, this video's simple solutions type, in tensors to deduce functionalities for my science idea and for my scifi story to have teleporting, ..
CHECK my community post, i say wormholes explain entanglement particle spin correlation by hyperspace (strnage) String Theory and Einstein crystalized time.. strings in time in hyperspace, [pre ER=EPR 2013, me aug. 3rd 2011 dated, but id thought it in 07ish watching PBS Nova special for The Elegant Universe] so now I know its called non local, so non local in our 3 Ds could be local in hyperspace.. basically String Theory dimensions in format of Tensors from Einstein.. I have my own theory, which I havent heard.. that the extra dimensions would be from the different fields are like own universes, we are a hyperverse of parellel universe (in just our membrane, String Theory, and other membranes are more hyperspace strings vibe in, The Nova special said attached to another n vibes in ours, I think and for scifi would use that the Strings are attached to our membrane and vibe [infinitely diffractional said, but that'd be to explain doubleslit experiment] in other membrane, and when the string is vibing in nothingness between branes, thats an "artificial field" or arti. brane, like QLED TVs' artificial atom aka quantum dots. Then, pop into the artificially field and then back into our brane aka universe, or different hyperspaces, and thats how u teleport, and the science I want to do, experiments, and for my scifi (Psy V Cy aka Psychics Vs Cyborgs PvC), is just advancing double slit experiment like 2022 Nobel in physics on non local universe entanglement stuff.
Chat gpt gave tensors and added 1 dimensions, but wasnt aware I needed multiple dimensions, and probably tensors in the matrix tensor grid, , because our theoretical hyperspace in parellel universes, and so like maybe .. well my first big breakthru, after ignoring schizophrenic voices finally I instantly thought the differnet fields are own universes, from that spacetime and magnetic fields seemed plausibly different Spaces since I was young, thats what i thought watchin the Nova string theory special 07ish. And round sept. 8th 2017 i thought theyre own spaces with own times, and the times' time like thing is gravity.. so the tensor for that.. like a 3+1 D tensor with each dimension input be a tensor of another 3d space.. i theorize would be how entanglement is possible.
Please answer and read this.