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.

****************************************************
YOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems!
****************************************************

****************************************************
Other Course Playlists:

***************************************************

► Want to learn math effectively? Check out my "Learning Math" Series:

►Want some cool math? Check out my "Cool Math" Series:

*****************************************************

*****************************************************

*****************************************************

This video was created by Dr. Trefor Bazett.

BECOME A MEMBER:

MATH BOOKS & MERCH I LOVE:

[lordfarquaadreal]

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

[loden]

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

[robbertcox]

Great explanation, your enthusiasm at the end is infectious

[leoliu]

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

[samedbey]

You are a gift sent from god to math enthusiasts. Thank you.

[rakoonberry]

Your videos are awesome. Keep up the good work!

[expansion_forever]

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

[김동완-jj]

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

[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!

[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

[spacepanda]

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

[lukenelson]

This was a great explanation, thank you!

[horsethif]

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

[bigseekersb]

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

[oxo]

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.