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Results tagged with asymptotics
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user 498835
Asymptotic behavior of functions, asymptotic series and related topics
46
votes
Accepted
A challenging (for me) limit calculation
This limit converges to $\frac{\sqrt3}2$. The idea is that $\sin(x) = x - \frac{x^3}6 + O(x^5)$, so we start with $\frac1{\sqrt n}$ and repeatedly subtract $\frac{x^3}6$. We can approximate this discr …
- 3,319
8
votes
Accepted
Series with the smallest number whose square is divisible by $n$
I couldn't find a reference, but (as noted in the OEIS page) if we have $k = a b^2$ with squarefree $a$ then $a(k) = ab$, so $$\begin{align*}
\sum_{k\leq x}\frac1{a(k)}
&= \sum_{a b^2 \leq x} \frac{\m …
- 3,319
0
votes
Expected sorting time of random permutation using random comparators
Here's a proof that $\mathbf{E}[X] = O(n^2 \log^2(n))$. Let's assume WLOG $n$ is a power of 2. Let's calculated the expected time until all values bigger than $\frac{n}2$ are in the second half (and a …
- 3,319