Transseries, nest-series, and other exotic series representations for tetration
#27
(12/15/2009, 12:46 PM)bo198214 Wrote: Hm, I have to look where the error is then.

It appears that the Faulhaber sum is not monotonic (which I assumed), i.e. we dont have that from \( f(x)<g(x) \) follows that \( \sum_{x} f(x) < \sum_{x} g(x) \). This puzzles me a bit, shouldnt one expect that from a sum operator?

But no, one shouldnt. The simplest example is perhaps \( f(x)=1 \) and \( g(x)=1+x^2 \) with the Faulhaber sums \( x \) and \( \frac{1}{3} x^{3} - \frac{1}{2} x^{2} + \frac{7}{6} x \); the latter cuts \( x \) at 0 because the first derivative there is different from 1.


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