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

Click For Summary
SUMMARY

The discussion focuses on deriving the series expansion for ln((x+1)/(x-1)) by utilizing the known series expansions for ln(x+1) and ln(1-x). The series for ln(x+1) is x - (x^2/2) + (x^3/3) - (x^4/4) and for ln(1-x) it is -x - (x^2/2) - (x^3/3) - (x^4/4). By subtracting the series for ln(1-x) from ln(x+1), the resulting series expansion for ln((x+1)/(x-1)) is 2x + (2/3)x^3 + (2/5)x^5 + (2/7)x^7.

PREREQUISITES
  • Understanding of Taylor series expansions
  • Familiarity with logarithmic functions
  • Knowledge of series manipulation techniques
  • Basic algebraic skills
NEXT STEPS
  • Study Taylor series for various functions
  • Learn about logarithmic identities and properties
  • Explore series convergence and divergence
  • Investigate advanced series manipulation techniques
USEFUL FOR

Mathematicians, students studying calculus, and anyone interested in series expansions and logarithmic functions will benefit from this discussion.

brendan
Messages
64
Reaction score
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
 
Physics news on Phys.org
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:
 
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
 

Similar threads

MHB How can I develop ln(x) into a series for x >= 1 in fluid dynamics?
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Series expansion of ##(1-cx)^{1/x}##
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Verifying Integration of ##\int_0^1 x^m \ln x \, \mathrm{d}x##
  • · Replies 1 ·
Replies
1
Views
3K
High School Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
  • · Replies 3 ·
Replies
3
Views
2K
High School Calculating shaded area: I'm getting discrepancies between methods
  • · Replies 3 ·
Replies
3
Views
3K
MHB Maclaurin Limit of lnx: \frac{1}{2}
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Taylor series of 1/ln(t+1) at t=0
  • · Replies 10 ·
Replies
10
Views
4K
MHB Finding Maclaurin series of a natural log function
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Method for finding a complex series?
  • · Replies 10 ·
Replies
10
Views
3K
MHB Can Series Expansion Prove the Relation Between Inverse Coth and ln(x+1)/(x-1)?
  • · Replies 3 ·
Replies
3
Views
2K