# Taylor’s Theorem and Taylor series

• Difficulty Level : Hard
• Last Updated : 03 May, 2020

Taylor’s theorem is used for the expansion of the infinite series such as etc. so that we can approximate the values of these functions or polynomials. Taylor’s theorem is used for approximation of k-time differentiable function.

Statement:
Let the (n-1)th derivative of i.e. be continuous in the nth derivative exist in and be a given positive integer. Then there exists at least one number lying between 0 and 1 such that:
…..

where and
Putting x=a+h or h=x-a we write equation as:
…..

Taylor’s remainders Rn after n terms due to:
1. Cauchy: we just put p=1 in the Taylor’s theorem to get
2. Lagrange: p=n gives

Taylor’s formula :
Using Lagrange’s remainder we get the Taylor’s formula:
…..
where
As n →∞ if R→0 then the last term of the formula becomes

Therefore the Taylor’s formula further reduces to

This formula is now used to give the infinite series expansion of f(x) about point a.

Example:
Obtain the Taylor’s series expansion of

Explanation:
According to the formula we have a= -1 here and f(x) is provided to us. First of all we need to calculate f(a) and then we calculate derivatives of f(x) at given point until it becomes zero.

Now we stop here as the next derivative will be zero. f^n(x)

=0 for n>5

Thus the Taylor series expansion of f(x) about x= -1 is:
…..

Substituting the values as calculated by us we get

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