Finding this function's series expansion

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Homework Help Overview

The discussion revolves around finding the series expansion of the function ln[x + (1+x²)¹/²], specifically exploring different methods to achieve this expansion.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the use of the MacLaurin series and express interest in alternative methods for finding the series expansion. One participant suggests taking the derivative of the function to simplify the process, while another mentions the connection to the arcsinh function and proposes integrating the series for its derivative.

Discussion Status

The discussion is active, with participants sharing insights and alternative approaches. There is acknowledgment of different methods being explored, but no consensus has been reached on a single preferred method.

Contextual Notes

Some participants are considering the implications of using derivatives and integrals in their approaches, while others are looking for additional mathematical tricks to enhance their problem-solving toolkit.

tamtam402
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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.
 
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tamtam402 said:

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 [itex]\displaystyle \ln\left(x+\sqrt{1+x^2}\right)\ .[/itex]

The resulting function looks much easier to work with.

Then integrate term by term to get your final result.
 
Last edited:
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!)
 
Thanks to you two!
 

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