Expand the series e^x sin^(-1)x – Apply Maclaurin series. Hello friends, today I’ll expand the series by using the Maclaurin series. Have a look!!

### Expand the series e^x sin^(-1)x

**Example**

Disclaimer: This is not my own example. I have chosen it from some book. I have also given the due reference at the end of the post.

So here is my example.

**Example **

According to Stroud and Booth (2013)*, “By the use of Maclaurin series, show that

Assuming the series for , obtain the expansion of , up to and including the term in . Hence show that, when is small, the graph of approximates to the parabola .”

**Solution**

Now here I have to show that

And I have already shown it in one of my earlier posts on the **Maclaurin series of **. So here I wouldn’t repeat it.

Next, I have to assume the series for . And that is also done in the **Maclaurin series for **. Thus the first task is to expand up to and including the term in .

**Step 1**

Now the series expansion of is

(1)

Also the series expansion of is

(2)

Next, I’ll expand the series .

So I’ll multiply equations (1) and (2) to get

Now I’ll expand it. First of all, I’ll multiply with . Then I’ll multiply with .

So that gives

Since I can get the expansion of up to and including the term , I won’t do further multiplication. Also, I’ll discard terms containing powers of higher than .

Thus the value of will be

Then I’ll simplify it. And that gives

So this means

Hence I can conclude that the series expansion of up to and including the term in is

And this is the answer to the second part of the example. Now I’ll do the last part of this example.

**Step 2**

As I can see from step 1, the series expansion of is When is very small, the terms from onwards will be extremely small. So we can neglect those terms from onwards.

Thus the value of will be

Now the standard form of a parabola is

If I compare it with the value of , I can see that and . Therefore is the equation of a parabola.

Hence I have proved the last bit of this example as well.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

## Leave a Reply