Finding this function's series expansion

[tamtam402]

Homework Statement

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

Homework Equations

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.

 

[SammyS]

Homework Statement

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

Homework Equations

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.

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.

 

[vela]

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!)

 

[tamtam402]

Thanks to you two!