# Proof of Taylor's series

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#### neptune

##### Member
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

##### Member
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

##### Well-Known Member
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

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

#### Ratchit

##### Well-Known Member
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.
http://www.dcs.warwick.ac.uk/people/academic/Steve.Russ/cs131/NOTE26.PDF

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