[AW] Update summary to new version of helpers

This commit is contained in:
2025-09-26 12:19:22 +02:00
parent 6ac6ee24cc
commit 33f034fdd1
16 changed files with 57 additions and 50 deletions

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@@ -74,7 +74,7 @@ With that, let's determine
\[
\E[\mathcal{X}] = \sum_{i = 1}^{n} \E[\mathcal{X}_i] = \sum_{i = 1}^{n} \frac{n}{n - i + 1} = n \cdot \sum_{i = 1}^{n} \frac{1}{i} = n \cdot H_n
\]
where $H_n := \sum_{i = 1}^{n} \frac{1}{i}$ is the $n$th harmonic number, which we know (from Analysis) is $H_n = \ln(n) +$\tco{1}, thus we have $\E[\mathcal{X}] = n \cdot \ln(n) +$\tco{n}.
where $H_n := \sum_{i = 1}^{n} \frac{1}{i}$ is the $n$th harmonic number, which we know (from Analysis) is $H_n = \ln(n) + \tco{1}$, thus we have $\E[\mathcal{X}] = n \cdot \ln(n) + \tco{n}$.
The idea of the transformation is to reverse the $(n - i + 1)$, so counting up instead of down, massively simplifying the sum and then extracting the $n$ and using the result of $H_n$ to fully simplify