Study of Taylor Series: Definition, Theorem, Applications, & Examples

**Publisher : Eliza Parker**

A Taylor series is a denoted function as an infinite sum of terms, where each term is a coefficient multiplied by the power of the independent variable. The series is named after the mathematician Brook Taylor and is a fundamental tool in calculus and mathematical analysis.

A mathematician Brook Taylor first presented the concept in the 18th century. The Taylor series is a dominant tool in mathematics and analysis that can be used to approximate functions, evaluate integrals, and explain differential equations.

The Taylor series can be used to estimate the value of a function at any point in its domain, and the correctness of the approximation depends on the number of positions involved in the series.

The concept of the Taylor series is an essential tool in modern mathematics and has led to numerous advances in analysis, physics, engineering, finance, number theory, and other fields. In this article, we will discuss the basic definition, theorem, application, advantages, and disadvantages.

**Taylor polynomial
series**

derives from the function's derivatives at a single point.**f(z)**

about a point **z = a**

is given by :**f(z) = f(a) + f'(a)*(z-a) + f''(a)*(z-a) ^2/2! + f'''(a)*(z-a) ^3/3! + ...**

Now,

**f'(a), f’’(a), f'''(a), ...**

represent the 1st, 2nd, 3rd, and higher derivatives of **f(z)**

calculated at **z=a**

, and the symbol **n!**

denotes the factorial of n.The Taylor theorem conditions that several appropriately smooth functions f(z) can be approached by its nth-degree Taylor polynomial series at a specific point “

**a**

”, with the residue term **Rn(z)**

assumed as:**f(z) = f(a) + f'(a)*(z-a)/1! + f''(a)*(z-a)**^{2}/2! + ... + fn(a)*(z-a) ^{n}/n! + R_{n}(z)

Where

**R**_{n}(z)

is the residue period, which is the same as the **n**^{th}-derivative

of f(z) calculated at particular point c between “**a**

” and “z”, multiplied by the **(z-c)**^{(n+1)}

divided by **(n+1)!**

which is **R**_{n}(z) = [f^{(n+1)} (c)*(z-c) ^{(n+1)}] / (n+1)!

The representation

**f**^{(n)}(z)

symbolizes the **n**^{th}

derivative of **f(z)**

concerning z.Disadvantages and Advantages :

Higher-order terms :

Example section :

**f(z) = sin(z)**

centered at **z=0**

:f(z) = sin(z)

f(0) = sin(0) = 0

f'(z) = cos(z)

f'(0) = cos (0) = 1

f''(z) = -sin(z)

f''(0) = -sin (0) = 0

f'''(z) = -cos(z)

f'''(0) = -cos (0) = -1

f''''(z) = sin(z)

f''''(0) = sin (0) = 0

...

Using this information, we can write the Taylor series for sin(z) centered at z=0 as:

sin(z) = z - z^3/3! + z^5/5! - z^7/7! + ...

The general formula for the Taylor series expansion of a function f(z) centered at

**z=a**

is :**f(z) = f(a) + f'(a)*(z-a)/1! + f''(a)*(z-a) ^2/2! + f'''(a)*(z-a) ^3/3! + ...**

Where

**f', f'', f’’’**

, represent the **f(z)**

, correspondingly.The above example of the Taylor polynomial can also be solved with the help of a

**Taylor series calculator**

to get rid of lengthy and time-consuming calculations.Conclusion :