Proof of Taylor's series

[neptune]

Hello folks,

I was studying Taylor's series,
Question 1: but i don't know how they got the formula for Taylor series

books always starts with the power series of

but why ?

Definition :" Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. " (source - wikipedia)
Question 2: what is k times differentiable.

 

[djsfantasi]

A function f(x) is k times differentiable if the differential equations f'(x), f''(x),...,f''...k times(x) all exist. (Sorry, need to brush up on Latex)

Take the function
f(x)=3*x^2

f'(x)=6*x
f''(x)=6
f'''(x)=0

Where I fall down is deciding if this is infinitely differentiable, or twice or thrice differentiable. Help?

 

[Ratchit]

Hello folks,

I was studying Taylor's series,
Question 1: but i don't know how they got the formula for Taylor series

books always starts with the power series of

but why ?

Definition :" Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. " (source - wikipedia)
Question 2: what is k times differentiable.

That theorem is presented and expounded upon extensively in just about any calculus textbook.

Ratch

 

[ESDR]

hello,
I have also studied the taylor theorem but don't understand the practical application where it is seen?

 

[Ratchit]

Hello folks,

I was studying Taylor's series,
Question 1: but i don't know how they got the formula for Taylor series

books always starts with the power series of

but why ?

Definition :" Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. " (source - wikipedia)
Question 2: what is k times differentiable.

Google is your friend.
**broken link removed**

 

[Ratchit]

hello,
I have also studied the taylor theorem but don't understand the practical application where it is seen?

Google is your friend.
https://math.stackexchange.com/ques...e-practical-applications-of-the-taylor-series