Introduction to Taylor series and Maclaurin Series

Taylor's and Maclaurin series expansion are methods of expanding a function about a signal point. It is an application of differential calculus that uses higher-order derivatives to approximate a function about a point. But it is only applicable if a function is infinitely differentiable. Let us learn more about Taylor's series and its application in mathematics.

Understanding the Taylor’s and Maclaurin Series

Taylor’s series is an application of derivative used to expand a function in the form of f(a+h). It writes the original function as higher-order derivatives to approximate the solution. Taylor’s series allows us to write a function in terms of a geometric series, giving important information about the function. For example, it calculates the value of a function at a point along with all its higher derivatives. 

By definition, the Taylor’s series expansion is stated as,

Taylor’s series is a function of the sum of infinite n terms of a function which can be expressed as the sum of n derivatives of the function at a single point. If a real or complex-valued function f(x) is infinitely differentiable at a point a, then Taylor’s series expansion will be represented as,

The above expansion gives the real-valued of the derivatives f(a), f’(a), f’’(a) up to nth order. n! denotes the factorials of real numbers in the denominators. If the point a is zero, then the above expansion will be called the Maclaurin series and will be reduced to,

Taylor’s Series Expansion Formula

Taylor's expansion is used to find the infinite sum of derivatives of a function at a single point. If a real or complex-valued function is infinitely differentiable at a real or complex point a, the Taylor’s series formula will be written as:

$f(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\cdots$

By using sigma notation, the Taylor’s series expansion can be written as,

Where, fn(a) represents the nth derivative of f(x) at the point a. Let us discuss how to use Taylor’s series in a step-by-step method.

How do you do a Taylor series step by step?

For a function f(x), To calculate Taylor’s expansion at the point a, the following steps can be used. 

1. Calculate first, second and third derivatives of f(x).
2. Find the value of f(x) and all of its derivatives at the point a.
3. Find the terms x-a, (x-a)2 etc according to the Taylor’s series formula. 
4. Substitute the values of all derivatives and x-a terms in the formula. 
5. Simplify the result by using a summation formula.

 Let’s understand how to calculate Taylor’s series of a function in the following example.

Taylor’s Series Example 

Find the Taylor’s series of the function f(x) = cos x at point x = 0.

Given that, 

$f(x)=\cos x$

To find Taylor’s series expansion, we need to calculate first few derivatives of cos x, these derivatives are,

$f'(x)=-\sin x$

$f''(x) = -\cos x$

$f'''(x) =\sin x$

$f''''(x)=\cos x$

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.

.

Now finding the values of all derivatives of f(x) at x = 0.

$f(0) = \cos 0 = 1$

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

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

$f'''(0) = \sin 0 = 0$

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

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.

.

Now substitute the values of derivatives in the Taylor’s series expansion formula, we get

$f(0)=f(0)+\frac{f'(0)}{1!}x+\frac{f''(a)}{2!}x^2+\frac{f'''(a)}{3!}x^3+\cdots$

$f(0)=1 + 0 + \frac{1}{2!}x + 0 + \frac{1}{4!} x^4 +\cdots$

Or, 

$f(0)=1 +\frac{x}{2!} + \frac{x^4}{4!}+\cdots$

Now by using the summation formula, we can generalized the expansion for cos x.

Maclaurin Series Formula

This series is named after the Scottish mathematician Colin Maclaurin. For a infinitely differentiable function f(x) up to nth order, the Maclaurin series can be expressed as:

Where a = 0. By using sigma notation, the Maclaurin’s series expansion can be written as,

$f(x)=\sum_{n=0}^{\infty}\frac{f^n(0)}{n!}x^n$

Where, fn(0) represents the nth derivative of f(x) at the point a=0. Let us understand how to calculate Maclaurin’s series expansion. 

How do you do a Maclaurin series step by step?

For a function f(x), To calculate Maclaurin’s expansion at the point a=0, the following steps can be used. 

1. Calculate first derivatives, second derivatives and third derivatives of f(x).
2. Find the value of f(x) and all of its derivatives at the point a=0.
3. Find the terms x, x2 etc according to the Maclaurin’s series formula. 
4. Substitute the values of all derivatives and xn terms in the formula. 
5. Simplify the result by using a summation formula.

 Let’s understand how to calculate Maclaurin’s series of a function in the following example.

Maclaurin’s Series Example 

Find the Maclaurin’s series of the function $f(x)=\cos x$ at point $x = 0$

Given that, 

$f(x) = \cos x$

To find Maclaurin’s series expansion, we need to calculate first few derivatives of cos x, these derivatives are,

$f'(x)=-\sin x$

$f''(x)=-\cos x$

$f'''(x)=\sin x$

$f''''(x)=\cos x$

.

.

.

Now finding the values of all derivatives of f(x) at x = 0.

$f(0)=\cos 0 = 1$

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

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

$f'''(0)=\sin 0=0$

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

.

.

.

Now substitute the values of derivatives in the Maclaurin’s series expansion formula, we get

$f(0)=f(0)+\frac{f'(0)}{1!}x+\frac{f''(a)}{2!}x^2+\frac{f'''(a)}{3!} x^3+\cdots$

$f(0)=1 + 0 + \frac{x}{2!} + 0 +\frac{x^4}{4!} + \cdots$

Or, 

$f(0) = 1 + \frac{x}{2!} + \frac{x^4}{4!} + \cdots$

Now by using the summation formula, we can generalized the expansion for cos x.

Difference between Taylor’s series and Maclaurin’s series 

The comparison between the Taylor’s and Maclaurin series can be easily analysed using the following difference table.

Conclusion

Taylor's series expansion is a method of expanding a function by using higher derivatives and the Maclaurin series is a special case of Taylor’s series. It is applicable when a function is infinitely differentiable at a single point. According to the above discussion, we can conclude that Taylor’s series approximate a function at point whereas the Maclaurin series approximates at zero. It has a wide range of applications in mathematics and other fields of science such as physics, economics, finance, engineering etc.