How to Calculate Taylor Series?

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The Taylor Series, which is sometimes called the McLaurin Series on special circumstances, is a popular power series for functions. The Taylor series of a function is simply a representation of the function infinite sums of terms which are expressed in the form of a derivative at a sine point.

This power series is very easy to determine as your only need to find the n + 1 terms to get the nth polynomial. For example, if you want to find the derivative of a function polynomial degree of 5, you will need to calculate the Taylor series of that function as the 6th sum of terms (n + 1).

The Taylor series polynomials are an approximation of a function whose accuracy gets better as n increases on a constant point of a.  when a = 0, then what you get will be the McLaurin Series. 

This article will discuss the formula of the Taylor Series and how to accurately use it to calculate the approximations of functions such that the derivate becomes a very good presentation of that function.

Taylor Series Formula

The Taylor Series of Function f(x), which is infinitely differentiable at a centered real or complex number denoted by  a is the power series;

F(x)= \frac{f(a)}{0!}(x-a)^{0} + \frac{f^{/}(a)}{1!} (x-a)^{2}+ \frac{f^{//}(a)}{2!}(x-a)^{2}+\frac{f^{///}(a)}{3!} (x-a)^{3}+...

The above series can be written as

Where  f^{n}(a) represents the nth derivative that is evaluated at the point of a

If a=0, then the Taylor series becomes a special power series known as the McLaurin Series.

The two formulas of the Taylor Series are the same. However, the first is expanded expansion of the former. It is the preferred form when calculating this power series as it helps students better substitute and et their answers.

How to Calculate the Taylor Series?

You have to know how to calculate derivatives using the product rule, quotient rule, and Chain Rule. Understanding the concept of factorial is also very important. 

Write out the formula. This is extremely important as it will help you get the answer more quickly

Expand the formula up to the fourth series as this is usually considered enough to predict the series limit

Determine the respective  f(a), f^{/} (a),f^{//} (a),f^{///} (a),  and f///a and substitute into the expanded formula

Simplify as much as possible to arrive at the Taylor Series

What is Needed to Calculate the Taylor Series?

The Taylor Series is generally very easy to calculate as all you need to do is follow the steps of calculation outlined above. However, it is important that you know certain mathematical notations and subjects to successfully get an accurate calculation. Firstly, the summation term simply shows that the series can attain a derivative up to infinity. However, most calculations stop at the fourth derivative as it already signifies the direction of the series. 

Secondly, you need to have a very good understanding of differentiation and its different calculation methods. There is the quotient rule which is used for differentiation functions and variables with divisions. The product rule is for calculating the multiplying functions. Thee is also the chain rule which is for computing complex functions.

Thirdly, knowing how to calculate factorials is also very important. However, you will most likely not need to calculate more than the 4! Knowing the calculation will help you understand several divisions when dealing with the Taylor series.

Example 1

Find the Taylor Series for e^{x}  with a centered point of a=0

Solution

The first step is to write out the formula 

The Taylor series formula is

The second step is to expand the formula

e^{x}=\frac{f^{0}(a)}{0!)}(x-a)^{0}+\frac{f^{/}(a)}{1!)}(x-a)^{1}+\frac{f^{//}(a)}{2!)}(x-a)^{2}+ \frac{f^{///}(a)}{3!)}(x-a)^{3}+...

The third step is to determine faand the respective derivatives f^{/}(a),f^{//}(a), f^{///}(a),

Clearly;

Since f(x) = e^{x}, f(a) =e^{a} where x = a

Since f^{/}(x) = e^{x}, f^{/}(a) =e^{a}

Since f^{//}(x) = e^{x}, f^{//}(a) =e^{a}

Since f^{///}(x) = e^{x}, f^{///}(a) =e^{a}

This is true because the derivative of an exponential with a coefficient of 1 and a positive x will always be the same.

Substituting the above into the formula will give 

Recall that the series is centered at a = o. so substitute o anywhere a is found in the formula

Obviously, any value or variable with a power of 0 will always be equal. So e^{0} =1

This Taylor series is known as a McLaurin series because a is centered at 0

Example 2

Substitute the derived values into the expanded formula

Conclusion 

From the above examples, you can see that Calculating the Taylor series is no more than just a sequence of steps and putting the right numbers and variables where they fit. It does not require many complexities on the surface. The only area where you may find it difficult is when you might need to find a complex derivative, which is an entirely different topic. 

Generally, when a=0, the Taylor series is much easier to calculate because a is naturally eliminated from the formula leading to a more straightforward answer. You can take on more examples of your choice with the same formula, and you will arrive at an accurate answer.

How to Calculate Taylor Series?

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