# Finding this function's series expansion

1. Jun 13, 2012

### tamtam402

1. The problem statement, all variables and given/known data

I'm trying to find the series expansion of ln[x + (1+x2)1/2].
2. Relevant equations

3. The attempt at a solution

I managed to find the MacLaurin series expansion by using the definition of MacLaurin series, which means I had to derive the function multiple times. However, I'd like to know if there's another way I could've used to find the series, which would be another mathematical trick I'd add into my arsenal.

2. Jun 13, 2012

### SammyS

Staff Emeritus
Sure:

Take the derivative of $\displaystyle \ln\left(x+\sqrt{1+x^2}\right)\ .$

The resulting function looks much easier to work with.

Then integrate term by term to get your final result.

Last edited: Jun 14, 2012
3. Jun 13, 2012

### vela

Staff Emeritus
That function is also known as arcsinh x. Try finding the series for the derivative of arcsinh and then integrate that result term by term.

(Beaten by mere seconds!)

4. Jun 14, 2012

### tamtam402

Thanks to you two!