Binomial Series interactive applet
by M. Bourne
Background
This interactive applet explores the concept of a binomial series approximating a function, which we met in the last section, 4. The Binomial Theorem.
At the end of that page, there are two examples, which asked:
Example 5: Using the binomial series, find the first four terms of the expansion `sqrt(1+x)`;
and
Example 6: Using the binomial series, find the first four terms of the expansion `sqrt(4+x^2)`.
In both examples, we needed to approximate a decimal value using the series we obtained.
Restricted domain
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.
The interactive applet
In the following applet, you can examine the two examples and see what the approximating functions look like (where there is one term up to the case where there is twenty terms.)
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 (between `-1 < x < 1`) when we take more terms
- Remove terms using the "`-`" button, and observe the simple terms in the approximation curve
Choose example:
`y=root3(1+x)`
`y=sqrt(4+x^2)`
Approximating series
Remove terms: Add terms:
Number of terms:
You can see where these terms are coming from in the solutions given for Examples 5 and 6 on this page: 4. The Binomial Theorem.
Related sections
You can see similar cases where infinite series are being used to approximate given curves here:
Taylor Series (background information)
Maclaurin Series (background information)
The above ones use calculus to construct the series, and generally do not have the restricted domain associated with the binomial series expansion.