Expert: Paul Klarreich - 3/13/2009

Question
I was fiddling around with some Taylor polynomials for certain functions
recently and I found that the more terms I used for ln(1 + x), (since trying
to do so for ln(x) blew up in my face what with ln(0) and all) the worse the
approximation became, which is contrary to the purpose of Taylor expansions
as I understand it. Is there some profound reason behind this odd result?
Also, I worked out the value for the cos(i) and sin(i), with the interesting
result that cos(i) is a real number (!) but within the geometrical interpretation
of complex numbers in the complex plane, numbers of the form a + bi can
be described with a real angle from the Re axis, but what in the world does
an imaginary angle mean?

Answer
Questioner:   Anthony Cordisco
Category:  Advanced Math
Private:  No
 
Subject:  McClaurin Series for ln(1 + x)

Question:  I was fiddling around with some Taylor polynomials for certain functions recently and I found that the more terms I used for ln(1 + x), (since trying to do so for ln(x) blew up in my face what with ln(0) and all) the worse the approximation became, which is contrary to the purpose of Taylor expansions as I understand it.

1. >> Do you think that the purpose of Taylor series is to have a good time?  You usually do, but that is beside the point.

Is there some profound reason behind this odd result?

2. >> YES.  (See part 6, below.)

3. >> Isn't  ln(1 + x) a pretty standard expansion, which you can find in most textbooks or reference books?

See http://en.wikipedia.org/wiki/Taylor_series

which converges for  -1 < x <= 1
..............................
Also, I worked out the value for the cos(i) and sin(i), with the interesting result that cos(i) is a real number (!) but within the geometrical interpretation of complex numbers in the complex plane, numbers of the form a + bi can be described with a real angle from the Re axis, but what in the world does an imaginary angle mean?

4. >> for complex numbers you define  
       e^(iz) + e^(-iz)
cos z = -----------------
              2
which you can derive from the definition:

e^z = cos z + i sin z

Then you put  z = i.

...................
6. >> As to the human interpretation of an imaginary angle, and the profound reason behind the series for ln(1+x), the best reference I can give you is the Star Trek episode: "The Trouble with Tribbles".  You might not fully understand after watching, but at least you'll have a good time.  I can't do better than that.


Keep on fiddling.