Maclaurin Series

This simple article shows how the Maclaurin series works, and how to write out the expansions. Although modern calculators are able to show the expansions, it is worth learning them for basic functions such as sin x and cos x.

The expansion above shows the general formula of the Maclaurin series. I have animated it to make it simpler for students to understand. The series is one of the easiest to understand in mathematics and very repetitive as you can imagine. If you are ever reading a scientific paper, you will find mathematicians making a meal of this one to make their work appear more complicated than it really is. I always have a chuckle when I see one of these dressed up.

I was just recently reading some of the original manuscripts of Madhava, and Srinivasa Ramanujan and the amazing thing I noticed was that they did not have fancy names for their formulas. They were simply looking for mathematical steps that would work every time. Moreover, they were relying heavily on intuition and foresight, which is actually very refreshing to see because that is the thinking process required to make progress in this field. Therefore, in this article I will avoid all the confusing jargon and get straight into the interesting stuff that works. Once you have understood the principle, you can call it whatever you like, get bogged down by terminology, and limits.

The Expansion

The series describes the steps required to convert any function f(x) into its equivalent series expansion. Many functions can be expressed as a series expansion, but not all. How will you know if a function has a series equivalent? If the function has derivatives then there is a good chance it may expand as a series. As long as the numerator part is not zero, it has a good chance, however, the only way is to have a go and find out. Usually, trigonometry functions such as sine, cosine, tangent, and natural log Ln may expand as a series. This is how digital calculators are able to give you the sine or cosine values of any angle.

All you have to do is to find the values of the bits in red and plug them into the series. The red bit are called the derivatives of the function and you simply find the derivative and set x to zero to find its value.

You may be wondering, “What is a derivative, and how do you find it?” That is the simple bit, because all you have to do is to differentiate the function, set x to zero in the differentiated equation and find its value, known as the coefficient. Once you have that, you simply plug it into the general formula.

Plugging the red bits into the Maclaurin general formula is the easy part. Sometimes differentiating a function to find its derivative can be the most challenging part. Therefore, if you have not learnt differentiation, then you will have to learn that first.

Prime Mark

Prime Mark

The prime mark notation is a very simple way to show how many times the function differentiates. It saves you from having to write d/dx multiple times, and even better saves paper and ink. When I was a whippersnapper, I used to write it on the mirror, because you could easily rub it off thus saving paper.

As you can see, the first term in the series is just the value of the function with x set to zero. In all the examples shown below the function requires differentiation first and then its value found by setting x to zero.

first derivative

This indicates the result of the first derivative.

second derivative

This indicates the result of the second derivative.

third derivative

This indicates the result of the third derivative.

If you have a function that differentiates many times, then there is a good chance that a series approximation may exist for it. Just calculate the values of the red bits and plug them into the Maclaurin series to give you the series expansion formula.

Providing the series converges, the more terms you have in your series expansion the more accurate its calculated output will be. Ideally, you want to have as many terms as possible to increase the accuracy.

The examples that I am showing here are from my 1980s homework exercise. I figured it might be useful to someone. It shows how to derive the series expansion formula for sine, cosine, ln, and some basic mathematical expressions to start from a good base.


Man first thought of this type of mathematics in India. Madhava thought of this in 1350. The Scottish mathematician Colin Maclaurin 1698 - 1746 did the most with this series. I have always thought that his work was the most practical.

This Article Continues...

Maclaurin Series
Maclaurin Series Binomial Theorem
Maclaurin Series for sin x
Maclaurin Series for cos x
Maclaurin Series tan x
Maclaurin Series for e^x
Maclaurin Series for ln(1+x)
Maclaurin sin 2x
Maclaurin Series cosh x
Maclaurin Series xsin x
Maclaurin Series sin pi x
Maclaurin Series ln x
Series Expansions
Maclaurin sin x^2 - Homework