Finding the power series expansion of this ln

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Homework Help Overview

The problem involves finding the series expansion of the function ln(x + sqrt(1+x²)). The subject area relates to power series and logarithmic functions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to manipulate the logarithmic expression into a suitable form for series expansion. They consider using a known series for ln(1+x) but are unsure how to apply it. Another participant mentions using the Maclaurin series definition to derive the expansion but questions the necessity of this approach. Hints are provided suggesting a substitution method to simplify the expression.

Discussion Status

The discussion includes attempts to clarify the approach to the problem, with hints offered that may guide the original poster toward a more efficient method. There is an acknowledgment of different strategies being explored, but no consensus has been reached on the best approach.

Contextual Notes

Participants note a desire to avoid re-deriving series expansions and express concerns about the adequacy of their solutions. There is a mention of using external tools for assistance, which raises questions about the learning process.

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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?
 
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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.
 
Hint:

[tex]ln(x+\sqrt{1+x^2})=ln(x)+ln(1+\frac{\sqrt{1+x^2}}{x})[/tex]
 
micromass said:
Hint:

[tex]ln(x+\sqrt{1+x^2})=ln(x)+ln(1+\frac{\sqrt{1+x^2}}{x})[/tex]

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

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