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Proof of Series ln(1+x)


Date: 11/15/2001 at 12:15:31
From: Colleen Torke
Subject: Proof of series ln(1+x)

I need to show that the series ln(1+x) equals x-x^2/2+x^3/3-x^4/4... 
and so on... whenever x is between -1 and 1. I really do not know 
where to start, but I need an elaborate proof of it.  

Thanks for the help.


Date: 11/15/2001 at 15:08:04
From: Doctor Jubal
Subject: Re: Proof of series ln(1+x)

Hi Colleen,

Thanks for writing Dr. Math.

To solve this problem, you need to find the Taylor series for ln(1+x) 
around x = 0.

The idea behind a Taylor series is that if two functions have the same 
value and the same slope at a point, then they're going to be pretty 
close to each other near that point. And if the second derivative is 
also the same, then the match is even better, and the third derivative 
is also the same, better still. If all the derivatives are the same, 
then they are the same function, for most intents and purposes 
(assuming the functions are well-behaved).

This idea is embodied in Taylor's theorem
        
        ___  (x - x0)^k    d^k f
 f(x) = \    -----------   ----- (x0)
        /__      k!        (dx)^k
        k=0   

Here x0 is the point you're writing the Taylor expansion around. The 
term on the right is the kth derivative of f, evaluated at x = x0.

If x0 = 0, this series is called a MacLaurin series and has a slightly 
simpler form:
 
        ___   x^k   d^k f
 f(x) = \   ------ ------- (0)
        /__    k!   (dx)^k
        k=0

So to find this series, what you need to do is

  (1) Figure out what all of the derviatives of ln(1+x) are. Since 
      there are an infinite number of them, you can save yourself some 
      work by just taking the first few derivatives and then looking 
      to see if there's some sort of pattern you can use to predict 
      what the rest of them are.

  (2) Evaluate each of those derivatives at x = 0

  (3) Plug these values into the general form of the MacLaurin series, 
      and simplify until you have the series you want to prove.

Does this help?  If you need any further explanation, don't hesitate 
to write back.

- Doctor Jubal, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus
High School Sequences, Series

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