Example 6.  Use Maclaurin series and verify the identity  .
Solution 6.

Using the results of Examples 4 and 5 we have the series representations for   and   .  

Enter the Maclaurin series expansion for   .  

Differentiate the Maclaurin series expansion for    term by term.  

Enter the Maclaurin series expansion for  .  

Compare the series for  .

It always helps to look at a few of the terms of the series to see what is happening.  

It is now easy to see that    .

We are done.

Aside.  We could have Mathematica compute some terms in the Maclaurin series expansions of     and   .  

Again, it is now easy to see that    .

(c) John H. Mathews 2004