# Taylor and Maclaurin Series interactive applet

by M. Bourne

## Background

This interactive applet allows you to explore the Taylor and Maclauring Series examples we met in the last two sections, Taylor Series and Maclaurin Series.

The example on the Taylor Series page asked about finding the Taylor Expansion of `f(x) = ln x` near `x = 10`.

On the Maclaurin Series page we had examples which involved finding the Maclaurin Series expansion for `f(x) = sin x`, `f(x) = e^x` and `f(x) = cos x`, as well as finding `pi` using a Maclaurin Series expansion of `f(x) = arctan x`.

## The interactive applet

In the following applet, you can examine what's going on in the above examples and see what the approximating functions look like (for one term up to the case where there are twenty terms.)

The **original function** is the darker green one, and the **approximating curve** is the lighter gray one.

### Things to do

- Choose the
**function**of interest using the radio buttons at the top **Add terms**using the "`+`" button, and observe how the approximation curve gets closer to the original curve as we take more terms**Remove terms**using the "`-`" button, and observe the simple terms in the approximation curve

Choose example:

#### Approximating series

Remove terms: Add terms:

**Number of terms:**

You can see where these terms are coming from in the solutions given for the examples on Taylor Series and Maclaurin Series.

## Related sections

You can see similar cases where infinite series are being used to approximate given curves here:

Binomial Series interactive applet, which is based on the Binomial Theorem, and doesn't use calculus. The binomial series only "works" in the region `-1 < x < 1`. That is, we can find a good approximating curve in that reqion, but outside the region, the approximation is usually poor; and

Fourier Series Graph Interactive, which is based on differential and integral calculus.