User:EGM6321.f12.team4.Rui.C/Homework5 R5.9(2)

Statement
Use Taylor series at x=0 (MacLaurin series) to derive the following expressions:

Solution (1)
The required expression is denoted as follows:

Expansion of arctan(x) can be written as:

It can be expanded as:

Then it is clear that:

So the original equation required to derive is not correct and it should be to derive the following Taylor expansion:

It is also easy see because the original equation (5.9.1.1) does not hold at x=0.

Then a generalized expression for F is:

Author and References

 * Solved and Typed by -- Jinchao Lu and Rui Che