this is for holding javascript data
Niclas Alexandersson edited solving the sum.tex
about 10 years ago
Commit id: be754452eb4728faad1ec5e7c0572722044170ad
deletions | additions
diff --git a/solving the sum.tex b/solving the sum.tex
index f78bd7d..dd68865 100644
--- a/solving the sum.tex
+++ b/solving the sum.tex
...
&= a + \sum_{p = 2}^{\sqrt{n}}b + \sum_{p = 2}^{\sqrt{n}}\left[c\sum_{m = p}^{\frac{n}{p}} 1\right]\\
&= a + b\sum_{p = 2}^{\sqrt{n}}1 + c\sum_{p = 2}^{\sqrt{n}}\left[\sum_{m = p}^{\frac{n}{p}}\right]\\
&= a + b(\sqrt{n} - 1) + c\sum_{p=2}^{\sqrt{n}}\left[\frac{n}{p}-p+1\right]\\
&= a + b(\sqrt{n} - 1) +
c\left(n\sum_{p=2}^{\sqrt{n}}\frac{1}{p}+\sum_{p=2}^{\sqrt{n}}[1-p]\right)\\ cn\sum_{p=2}^{\sqrt{n}}\frac{1}{p}+c\sum_{p=2}^{\sqrt{n}}[1-p]\\
&= a + b(\sqrt{n} - 1) + cn\left(\sum_{p=1}^{\sqrt{n}}\frac{1}{p} - \sum_{p=1}^{1}\frac{1}{p}\right)\\
&= a + b(\sqrt{n} - 1) + cn(H_{\sqrt{n}} - 1)\\
&= a - b + b\sqrt{n} + cnH_{\sqrt{n}} - cn