Finding the power series expansion of this ln

[tamtam402]

Homework Statement

Find the series expansion of ln(x + sqrt(1+x²))

Homework Equations

ln(1+x) = x - x² /2 + x³/3 - x⁴/4 + ...

The Attempt at a Solution

I don't know how to solve this. If it was ln(1+f(x) ) I know I could substitute the x's for f(x) in the ln(1+x) series expansion, but I don't know what to do since the ln isn't in the correct form. Am I supposed to rederive the series from scratch?

 

[tamtam402]

Well I solved the whole problem by using the definition of Maclaurin series to find the expansion "from scratch". I used Wolfram Alpha to solve the multiple derivatives necessary, but I'm not satisfied with my answer. I feel like there was a way to solve this by using some kind of substitution, since I managed to solve all the other problems in Mary L Boas book without having to rederive the whole series.

I'm trying to learn the math tools necessary to self-study undergraduate E&M. I'm an electrical engineering student that would like to know the fundations of my engineering branch :D I feel bad for "cheating" on this problem.

 

[micromass]

Hint:

ln(x+\sqrt{1+x^2})=ln(x)+ln(1+\frac{\sqrt{1+x^2}}{x})

 

[tamtam402]

Hint:

ln(x+\sqrt{1+x^2})=ln(x)+ln(1+\frac{\sqrt{1+x^2}}{x})

Wow, I wouldn't have found that by myself. That's a very clever trick, thank you!