The McLaurin series is a popular power series developed from the popular Taylor Series. The McLaurin Series is considered a special case of the Taylor series because the constant point on which the derivative of the latter is being carried out is equal to zero.
Of course, you must have first studied what the Taylor series actually is before you can successfully move to the McLaurin series.
The Taylor series is used to approximate a function’s behaviour and estimate what the function would look like when full analysis and calculation are impossible. The Taylor series works with the summing of the consecutive derivatives of a function. The number of times the function derivative will need to be found is denoted as n, while the constant at which that function is being derived is denoted as .
can be anything. However, whenever it is equal to zero, we get what is known as the Mclaurin Series
This article focuses on the Mclaurin series and how it approximates a function behaviour with a similar pattern to the Taylor series. It is not difficult and is easily understandable. Each term and calculation will be explained step by step to aid understanding.
What is the McLaurin Series?
The McLaurin Series is a special form of the Taylor Series whose sum of infinite derivatives is considered at a point zero.
It is denoted as
The above formula is derived from the Taylor series, and more exposure to the two series connection will be considered in the next subheading.
Taylor Series and McLaurin Series connection
The Taylor series is one of the most effective power series that approximate the function, F(x). It is infinitely differentiable at a centered number denoted by a which could be a real or complex number.
It is denoted as
Now, if a = 0, then the Taylor series will become the McLaurin series. Actually, that will make the McLaurin Series.
As such, we can say the McLaurin Series is
How to Calculate the McLaurin Series?
The method of computing the McLaurin series is the same as the Taylor Series. The only major difference is the fact that the McLaurin Series is much easier to compute compared to the Taylor series, as you really do not have to stress much about finding
Step 1: Ensure that the Power series question has a constant value equal to zero. i.e a = 0
Step 2: find the equation up to the fourth derivative, which means n = 4
Step 3: solve the Mclaurin Series with the formula.
Examples of The McLaurin Series Being Used as an Approximation Tool
Example 1
Find the power series for the formula f(x) = ex at a = 0
Solution
Clearly, since the power series approximation has to have = 0, then the McLaurin series is a good choice to go for.
The formula is given as:
Where
F(x) = ex
Now we will expand the formula up to the fourth derivative so that we have an idea of the approximation behaviour
will become
The factorials will be expanded next
Therefore, the McLaurin Series is
Example 2
Findthepowerseries or sinx at a = 0
Solution
With the above information
So eliminating the parts that are automatically equal to zero will leave us with the equation below
Example 3
Calculate the McLaurin Series for the function F(x) = x4
Solution
For the above problem, it was specified that the approximation method required is the McLaurin series. As such, using the McLaurin formula is the first step to take in this case.
The formula is given as follows:
Where
F(x) = x4
As such, we will have to follow the same steps as the first example
Now since f(x) = X4, then f(0) = 04 =0
as such, the McLaurin series of X4 = 0
Now the answer gotten in example 2 give a very important detail about the McLaurin Series. When = 0, the approximation of most unction will tend towards zero. Only a few functions, when approximated via the McLaurin series will give a non-zero answer.
The first example is one of those few functions that stand out. The function in the example is what is known as the exponential function and is the most common function considered in McLaurin Series. It arrives so much in mathematical questions that it is almost okay to expect it most of the time. You can memorize it if you have a test that deals with the McLaurin series. It can help guide your answer and ensure you always end up with the correct solutions.
Just like the Exponential function, the McLaurin Series have other common function that students can actually compute using the same pattern established.
Conclusion
The McLaurin series follows a very simple system that you can understand without breaking much sweat. It is straight to the point and even easier to calculate than the Taylor series as the constant does not exist. i.e = 0. This piece has fully considered the series and how it can be quite different from the Taylor series and other Power series using examples. Clearly, all the examples follow the same method, and you must also stick to that pattern for all your McLaurin Calculations as that is the best way to get your answer with very little confusion
https://scholar.harvard.edu/files/adaemi/files/worksheet19.pdf
