en.wikipedia.org/wiki/Riemann_series_theorem
1 correction found
ln ( 2 r )
The correct limit is ln(2√r), not ln(2r). The square root is essential.
Full reasoning
The preceding asymptotic calculation for the alternating harmonic series gives
[
S_n=\ln 2 + \tfrac12\ln!\left(\frac{p_n}{q_n}\right)+o(1),
]
where (p_n) and (q_n) are the numbers of positive and negative terms among the first (n) terms of the rearrangement. If (p_n/q_n \to r>0), then
[
S_n \to \ln 2 + \tfrac12\ln r = \ln(2\sqrt r).
]
So the plain-text claim ln(2r) is mathematically incorrect; it is missing the square root. The display formula shown immediately after it already gives the correct expression (\ln(2\sqrt r)).
You can also see this from the density form of the formula: if the limiting fraction of positive terms is (\alpha), then the sum is (\ln 2 + \tfrac12\ln(\alpha/(1-\alpha))). Since (r=\alpha/(1-\alpha)), that becomes (\ln(2\sqrt r)), not (\ln(2r)).
2 sources
- Riemann series theorem - Wikipedia
Suppose now that ... the ratio p_n/q_n ... tends to a positive limit r. Then, the sum of such a rearrangement will be ... \ln\left(2\sqrt{r}\right).
- Same Numbers, Different Sums: Rearranging the Alternating Harmonic Series - Elite Math Program
The rearranged series then converges, and its sum is S = ln 2 + 1/2 ln( α/(1−α) ). ... Since p/N → α and q/N → 1 − α, their ratio tends to α/(1 − α). This proves the formula.