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What is Wrong With this Idea?

  1. Oct 18, 2012 #1
    If we set out to prove the irrationality of the natural logarithm of π (pie), by writing out the Taylor series centered at zero for the function y=π^x, with x=1, we have:

    π=1+Sum(ln^k(π)/k!) from k=1 to infinity.

    Since we know π is irrational, then ln(π) must be irrational or otherwise π=(a+b)/b

    For integer a and b, why is this not correct?
     
  2. jcsd
  3. Oct 18, 2012 #2

    Because any real number is the limit of a rational sequence, and series play this game, too. For example, we know the number [itex]\,e\,[/itex] is irrational, yet

    $$e=\sum_{n=1}^\infty\frac{1}{k!}$$

    so, according to your idea, we'd get that [itex]\,1\,[/itex] is irrational...

    DonAntonio
     
  4. Oct 19, 2012 #3
    Aw of course, i see now, thanks.
     
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