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Taylor Series Formula


A Taylor series is a series expansion of a function at a single point. When the Taylor series is centered with zero, then that particular series is called as Maclaurin series.  A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. A Taylor series is a series expansion of a function about a point. The Taylor series of a function is the limit of that function's Taylor polynomials, provided that the limit exists. A function that is equal to its Taylor series in an open interval is known as an analytic function in that interval.

A Taylor series, which is an expansion of a real function f(x) at single point x = a is given as the formula stated below :

Taylor Series Formula

or you can make this formula simple and more compact by using the sigma notion
Formula for Taylor Series
here n! represents the factorial of 'n' and ƒ (n)(a) represents the nth derivative of 'ƒ' calculate at a single point 'a'.

Related Calculators
Calculate Taylor Series Series Calculator
Arithmetic Series Calculator Geometric Series Calculator
 

Taylor Series Problems

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Below are the problems on Taylor series :

Solved Examples

Question 1: Find the Taylor Series for the function f(x) = ex at about x = 0 ?

Solution:
 
Given function: f(x) = ex
f(n)(0) = e0 = 1; n = 0, 1, 2, 3....
f(n)(x) = ex ; n = 0, 1, 2, 3....
Taylor series for the function f(x) at about x = a is given by $\sum_{n = 0}^{\infty}$ $\frac{f^{n}a}{n!}$ $(x - a)^{n}$

So, for the function f(x) = ex = $\sum_{n = 0}^{\infty}$ $\frac{x^{n}}{n!}$
 

Question 2: Find the taylor series expansion for the function f(x) = e-x at about x = -3 ?

Solution:
 
Given function: f(x) = ex
f(n)(-3) = e-3 = 1; n = 0, 1, 2, 3....
f(n)(x) = ex ; n = 0, 1, 2, 3....

Taylor series for the function f(x) at about x = a is given by $\sum_{n = 0}^{\infty}$ $\frac{f^{n}a}{n!}$ $(x - a)^{n}$

So, for the function f(x) = ex = $\sum_{n = 0}^{\infty}$ $\frac{e^{-3}}{n!}$ $(x + 3)^{n}$
 

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