Talk:Convergence of P-Series

Lemma

I feel the lemma is important enough to be a theorem of its own right. And if it gets its own page, it might as well be extended to say that if $0 < x \le 1$ then the improper integral diverges. It seems unfair to label him a lemma. --GFauxPas 07:23, 5 February 2012 (EST)

I'm sorry, what lemma? --prime mover 09:38, 5 February 2012 (EST)

err sorry,

$\displaystyle \int_1^{\to \infty} \frac {\d t} {t^x}$ converges for $x > 1$. --GFauxPas 09:40, 5 February 2012 (EST)

Okay, can think about it ... --prime mover 09:47, 5 February 2012 (EST)

Would it be a good idea to somehow emphasize that this proof only shows absolute divergence of the complex series on 0<Re(p)<1? The title emphasizes that for convergence, but it may not be as clear on divergence. The absolute divergence is not nearly as interesting as whether the series converges conditionally over that interval. That question is not relevant for the real series, but is quite relevant for the complex series.

Why is it necessary to redo the integral test? Isn't the fact that the absolute value of the complex terms is the same as the terms of the real series sufficient to prove both conclusions with a simple comparison test?