Maclaurin Series cosh x

Using the general expansion formula, find the Maclaurin expansion of cosh x. This is an easy one to perform as the derivative of cosh x is sinh x, and the derivative of sinh x is cosh x. Therefore, it is a simple matter of finding the highlighted bits and plugging them into the above equation.

We know that cosh 0 = 1.

We know that sinh 0 = 0.

As you can see the derivatives of sinh and cosh are very easy to find. This is an ideal question for beginners to learn how to use the Maclaurin series.

As you can see, it simply consists of ones and zeros.

At this point, we can determine that there is a repeating pattern.

I wonder if this could be used for binary encoding. I must try it out one day.

At this stage we plug the values of the derivatives when x = 0 into the equation shown above. The terms with the zeros cancel out obviously and it simplifies as shown below.

Here is the Maclaurin series expansion of cosh x. It is amazingly, simple to derive and a marvel to think that a series expansion could express a trigonometry function.