Differentiating a power series to find a new power series

 Objective: Differentiate a power series to find a new power series

We showed in an earlier post that it's  possible to differentiate a power series term by term to create a new power series. In the following example we are going to differentiate the binomial series term by term to find a new binomial  series.

Example

Solution

There are 2 ways to solve this problem. First, we can differentiate term by term the first expression to find the binomial series of the second expression. Another easier way is to notice that the second expression is almost the derivative of the first one.

Differentiating the first expression, we have:

We deduct that:

Let's calculate d/dx⎷1+x:

The binomial series for ⎷1+x: is given by:

Practice