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Results tagged with asymptotics
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user 143
Asymptotic behavior of functions, asymptotic series and related topics
5
votes
Asymptotics of the number of compositions whose summands are the divisors of a number?
The number of compositions of n with all parts equal to 1 or 2, for example, is the (n+1)st Fibonacci number, which grows like φn where φ = (1+√5)/2.
The number of compositions of 2p for p a prime, w …
- 14k
6
votes
Asymptotics of q-Catalan numbers
It's not hard to compute numerical values. If you do this, in the regime $0 < q < 1$ it looks like $C_n$ grows exponentially, i. e. $C_n \sim \alpha_q \beta_q^n$ for some constants $\alpha_q$ and $\b …
- 14k
18
votes
Accepted
Estimate for tail of power series of exponential function?
Let's instead consider the sum
$$ \sum_{k = A + C \sqrt{A}}^\infty {e^{-A} A^k \over k!} $$
which of course differs from yours just by a factor of $e^{-A}$.
Then this sum is the probability that a P …
- 14k
4
votes
Accepted
Asymptotics of a hypergeometric series/Taylor series coefficient.
Okay, you want the asymptotics of
[z^n] 1/((1-2z) sqrt(1-4z)). …
- 14k
15
votes
Accepted
spiral of Theodorus
Here's a sketch of a proof that the constant you want exists, and how to find it.
Let
$$
f(n) = \arctan(1) + \arctan(1/\sqrt{2}) + \arctan(1/\sqrt{3}) + \ldots + \arctan(1/\sqrt{n}).
$$
You want to …
- 14k
10
votes
Asymptotics of a Bernoulli-number-like function
I suspect the limit doesn't exist, believe it or not. It seems like it should! But plotting $f(n,2)$ for n from, say, 30 to 2000, it looks like we have
$$ f(n) = C + \omega(n) + o(1) $$
where $C \ …
- 14k
7
votes
Accepted
Asymptotics for the number of partitions of $n$ into odd prime parts
Flajolet and Sedgewick, Analytic Combinatorics (link goes to free, legal downloadable PDF of book), section VIII.6 treats the asymptotics of various types of partitions. …
- 14k