Is the Taylor Series Method Valid for Proving the Irrationality of ln(π)?

[Aspiring]

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?

 

[DonAntonio]

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?

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

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

so, according to your idea, we'd get that $\,1\,$ is irrational...

DonAntonio

 

[Aspiring]

Aw of course, i see now, thanks.