Search Results

I get a slightly different answer every time I look at it, so I hope the following is ok. I won't dot every last "i" in the error analysis. Let $F(t)=\sum_{i=0}^\infty a_i(t)$, where $a_i(t)=t^{2^i}2 …

(CORRECTED EDITION) By mucking around with expansions like Igor suggested, I found $$ j \approx J(n,k)= en^{1/2} - \tfrac14\ln(n) -\tfrac12\ln(2\pi k)-\tfrac14 e^2-\tfrac12. $$ It seems good when …

(Edited after Christian's comments.) For $0\le i\le n^{4/7}$, $$n!/(n-i)! = n^i \exp(-i(i-1)/(2n)+O(i^3/n^2)).$$ Approximate the sum for that range by the corresponding integral (a gaussian with the …

The approximation coming from the normal approximation of the binomial distribution is $$ k = \sqrt{-2n\ln(1-\epsilon)}.$$ To get more terms write $\Delta=-\ln(1-\epsilon)$, then by expanding $-\ln(1- …

There is a chapter in Knuth and Greene, Mathematics for the analysis of algorithms, that explains how to estimate this type of thing. If $B$ is fixed and $n\to\infty$, the central limit theorem might …

I'm guessing you can approximate it with the differential equation $$ x'(n) = -\log(x(n)). $$ The solution to this satisfies some equation involving the exponential integral special function, namely $ …

Many unsolved asymptotics problems can be written as inclusion-exclusion sums. There is no general method for solving them. …

EDITED: As pointed out by Anthony and John, my 2am solution was anything but. In summary, the conjecture is TRUE for $C$ smaller than approximately 0.6880137 and false for larger $C$. The exact value …

The problem statement exactly corresponds to the definition of the hypergeometric distribution. With this key-phrase in hand, it is easy to locate an extensive literature. Start with wikipedia for ba …

Since the vast majority graphs has trivial groups, and this is even more true (to exponential precision) for graphs with likely numbers of edges, the answer will be the same if you consider labelled g …

I don't know where this is given in detail, but the method is the same as that used to obtain the distribution of the maximum degree of a random graph with $p=1/2$. In this range, the fact that the d …

Here is an elementary approach, which shows how to find the nature of $nf_n(t)$ as $n\to\infty$. But I'm not going to bound error terms or such so this remains an outline until those details are fill …

The OP is unclear about the domain of $z$ required. If only real $z\to\infty$ is of interest, just realise that the terms near $k=\sqrt z$ dominate the rest and their shape is close to a normal densi …

Since you say that you only want the first few terms, one way you can do this type of thing is by making a contraction mapping. As $x\to\infty$, inspection shows $y\sim x$, so rewrite the equation as …

The limit is at least $2/\sqrt 3\approx 1.1547$. Write $\alpha=1/\sqrt 3$. If any strip length $\alpha n$ or more is used more than once, then obviously area $2\alpha$ can be chosen. On the other han …