Can I Combine the Series for ln(x+1) and ln(x-1) to Expand ln((x+1)/(x-1))?

In summary, the conversation discusses working out a series expansion for ln ((x+1)/(x-1)) and the potential use of existing series expansions for ln(x+1) and ln(1-x). The possibility of subtracting these expansions is raised, and the correct expansion for ln(1-x) is provided for reference.
  • #1
brendan
65
0
Hi all,
I am trying to work out a series expansion for ln ((x+1)/(x-1)).


I have got the series expansion for ln(x+1) ie x- (x^2/2) + (x^3/3) - (x^4/4) ...

and for ln(x-1) -x- (x^2/2) - (x^3/3) - (x^4/4) ...

Can I tie these two together to get the series for ln ((x+1)/(x-1)).

Or do I treat this as a new series.

regards
Brendan
 
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  • #2
Welcome to PF!

brendan said:
I am trying to work out a series expansion for ln ((x+1)/(x-1)).

I have got the series expansion for ln(x+1) ie x- (x^2/2) + (x^3/3) - (x^4/4) ...

and for ln(x-1) -x- (x^2/2) - (x^3/3) - (x^4/4) ...

Can I tie these two together to get the series for ln ((x+1)/(x-1)).

Or do I treat this as a new series.

Hi Brendan! Welcome to PF! :smile:

Yes, you can just subtract them …

but do you really mean ln(x-1) … at x = 0, that would be ln(-1) … do you mean ln(1-x)? :smile:
 
  • #3
Thanks a lot I did mean ln(1-x).

Heres the expansion

2x + (2/3)x^3 + (2/5)x^5+ (2/7)x^7

Once again thanks
 

What is a Maclaurin series expansion?

A Maclaurin series expansion is a mathematical concept used to represent a function as an infinite sum of terms. It is named after Scottish mathematician Colin Maclaurin and is a special case of a Taylor series expansion, where the expansion is centered at x=0.

Why is the Maclaurin series expansion useful?

The Maclaurin series expansion allows us to approximate a complicated function with a simpler one, making it easier to analyze and solve problems. It also helps us to find the values of functions at any point, even if the function is not defined at that point.

What is the formula for a Maclaurin series expansion?

The general formula for a Maclaurin series expansion is: f(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + ... + (f^(n)(0)/n!)x^n. This formula can be used to find the coefficients of the expansion for a given function.

What is the difference between a Maclaurin series expansion and a Taylor series expansion?

A Maclaurin series expansion is a special case of a Taylor series expansion, where the expansion is centered at x=0. A Taylor series expansion can be centered at any point, while a Maclaurin series expansion is specifically centered at x=0. Additionally, a Taylor series expansion includes higher order terms, while a Maclaurin series expansion only includes terms up to the nth derivative.

What are some applications of the Maclaurin series expansion?

The Maclaurin series expansion has many applications in mathematics, physics, and engineering. It is commonly used to approximate functions in calculus, solve differential equations, and analyze the behavior of physical systems. It is also used in signal processing, statistics, and finance to model and analyze data.

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