Why Taylor Series actually work: The Taylor Inequality

A power series for a function is only as good as its remainder. Thankfully, we have an incredibly powerful result for Taylor Series, namely that the remainders are "well controlled" by the Taylor Inequality. In examples like e^x this means that the remainder goes to zero for all values of x as n goes to infinity. That is, no matter how accurate you need me to be, I can take enough terms in my Taylor polynomial and ensure that level of accuracy.
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[@loden5677]

This is helping me understand the theorem, especially visualizing it. Thank you!

[@lordfarquaadreal9156]

thank you for explaining the "why though" that always burns in the back of my mind when learning. you're the best!

[@quant-prep2843]

this guy is a god sent

[@leoliu7492]

An excellent explanation on taylor's inequality. Thank you!

[@robbertcox6170]

Great explanation, your enthusiasm at the end is infectious

[@rakoonberry7879]

Your videos are awesome. Keep up the good work!

[@user-fh7yk8nm6s]

Your explanation is always very helful in answering why I learn.

[@expansion_forever]

You are so good. Excellent teaching. Please continue making such good videos. By the way Happy Holi.

[@tonyhaddad1394]

Amazing job , respect

[@ericperlroth8830]

Amazing explanation!

[@artofpsyi5383]

Understood! Thank you

[@tejeshwarreddy3041]

Great one mann♥️

[@lukenelson9727]

This was a great explanation, thank you!

[@shawnpee1605]

Amazing video, thank you!

[@DrTrefor]

Glad you liked it!

[@maxderoma]

Ciao Trefor, I need your help , seriously: do you know anybody of your caliber teaching physics on youtube ? I am having some difficukties with subject and can't find a teacher I like - probably I'm to used to your wonderful maths classes that I simply can't settle for anything with lower standard! Again you are the best!

[@mathsbro806]

U r a blessing!

[@vladm.6859]

great vid

[@Mohammadoudat]

Gold

[@spacepanda3375]

Hello, professor. Can you explain why are you omitting -a term for interval in example with f(x) = e^x ?

[@anshumanagrawal346]

Because we are taking the power series centered at 0, so a = 0 in this case

[@hosopc3532]

Great Video!

[@DrTrefor]

Glad you enjoyed it

[@Wise_Man_on_YouTube]

Professor, please tell me which will yield me a better result for a function evaluated at a point, Runge kutta method or Taylor series..
Your response will be appreciated.
Thanks

[@balazsszalai41]

I don't know, if anyone needs the answer for this now, but I think it's an interesting question.
I would say it mainly depends on how far are you from the point, where you know the value of the function. But it also depends on the function itself.
From the Lagrangian form of the remainder you can see how well it converges the taylor polynomial to the actual function.
The Runge-Kutta method will have an error proportional to the size of the step (for example the RK4 will have a 16 times smaller error for 2 times smaller step).
So you can see approximately which of the methods work better for the given case.
(If I'm wrong, someone can correct me)

[@General12th]

Nifty!

[@pewpewhuang4162]

what M, and d at 5:10 mean?

[@GhostyOcean]

M is an upper bound (possibly the LEAST upper bound) for the given derivative, and d represents the distance from your starting point.

[@horsethi3f]

The sigma notation for the Taylor series, shouldn't 'i' start at 0?

[@DrTrefor]

It is just an index, so as long a you are consistent it doesn't matter

[@bigseekersb]

Seems like all the good male calculus tutors like plaid shirts.

[@DrTrefor]

FACT

[@oxo010]

I replayed many sections to hear what was said as your voice trailed off at the end of a sentence, then gave up. Work on keeping your clear, and not speeding up, as you finish each sentence.

[@thechickenknight3062]

i can understand him perfectly. Try slowing down the video if you need. Hope this helps!