All Questions
Tagged with sequences-and-series asymptotics
779
questions
3
votes
2
answers
97
views
$\sum_{n=0}^{+\infty} \frac{x^n}{1+nx} \stackrel{x \to 1^-}{\sim} -\ln(1-x)$
How to prove that $\sum_{n=0}^{+\infty} \frac{x^n}{1+nx} \stackrel{x \to 1^-}{\sim} -\ln(1-x)$ ?
First try : I tried to compare the sum with the integral $\int_0^{+\infty} \frac{x^t}{1+tx}\mathrm{d}t$....
- 189
1
vote
0
answers
48
views
Informations about a sequence from tail behaviour
Suppose $\{c_n\}_n$ is a sequence of non negative reals. We have the following three informations about it.
(a) $\sum_{k \ge n}c_k \sim \frac{1}{2n}$
(b) $\sum_{k=2}^n \frac{kc_k}{\log n} \to \frac{1}{...
- 825
0
votes
0
answers
69
views
Landau Notation Problem
I have this function
$$ K_{n} = \int_{1}^{+\infty}\frac{1}{(1+t^2)^n}dt$$ $$ \text{Let }t\geq1,t^2+1\geq1+t\Leftrightarrow\frac{1}{1+t^2}\leq\frac{1}{1+t} \text{ and for } n \in {\mathbb{N^{*}}} : \...
2
votes
2
answers
180
views
Does the rate $\log n$ imply "almost harmonic"?
Let $\{c_k\}$ be a decreasing positive sequence such that $\sum_{k=1}^n c_k \sim \log n$. Does it say $c_k=O(1/k)?$
I have found a reference where it says that the converse is true. I tried to tackle ...
- 825
-1
votes
2
answers
63
views
Upper bound on the finite sum of $\sum_j x^j/j!$ [closed]
How can I derive an upper bound on the following finite summation,
\begin{equation}
S = \sum_{j=1}^k \frac{x^j}{j!},
\end{equation} where $0 < x$, in terms of $x$ and $k$ (it's perfectly fine to ...
- 23
1
vote
1
answer
56
views
Upper bound for $ \prod_{i=0}^{k^2} \left(1 + \frac{n-1}{k+(n-1)i}\right) \left(1 - \frac{1}{k+(n-1)i}\right)^{n-1}$
I am working on finding a good upper bound for the following product:
$ \prod_{i=0}^{k^2} \left(1 + \frac{n-1}{k+(n-1)i}\right) \left(1 - \frac{1}{k+(n-1)i}\right)^{n-1} $
With the following ...
- 85
0
votes
0
answers
36
views
Asymptotic expansions of sums via Cauchy's integral formula and Watson's lemma
I am looking for references that deal with the asymptotic expansions of sums of the form
$$s(n)=\sum_{k=0}^n g(n,k)$$
using the (or similar to) following method.
We have the generating function
$$f(z)=...
- 2,259
0
votes
1
answer
71
views
If the sum $\sum_{n=1}^\infty n a_n$ converges for positive $a_n$, what can we say about the sequence $a_n$? [closed]
Let $(a_n)$ be a sequence of positive numbers and assume that
$$\sum_{n=1}^\infty n a_n < \infty.$$
What can we then say about $a_n$? There are a few obvious things; $a_n\to 0$ and the series $\...
- 376
2
votes
1
answer
175
views
Explicit translation of Ramanujan’s condition for the Selberg Class?
According to many sites such as Wikipedia, the Ramanujan condition of the Selberg class can be stated as
$$\boxed{\forall\epsilon>0:a_n\ll_\epsilon n^\epsilon}$$
My question is: what exactly does ...
- 310
4
votes
1
answer
178
views
Do iterated factorials or iterated exponents grow faster?
This is purely out of curiosity and I'm not quite at the point in calculus where I know how to prove either for myself...
Given $$n^{n^{n^{n^n}}}$$ and $$(((n!)!)!)!$$
As n approaches infinity, which ...
- 86
1
vote
1
answer
60
views
Asymptotic expansion of an integral by expanding series first
Question: as $s\to\infty$ use the substitution $u = {\rm e}^{-x}\,$ to obtain the first 2 terms in asymptotic expansion in the integral:
$$
\operatorname{B}\left(s,t\right) =
\int_{0}^{1}u^{s - 1}\,\,\...
- 305
1
vote
1
answer
59
views
Quotient of two asymptotic expansions
I try to derive the asymptotic expansion$$a_n=\frac{I(n+1)}{I(n)}=\sqrt{n}+\dfrac{1}{2}-\dfrac{1}{8\sqrt{n}}+o\left(\dfrac{1}{\sqrt{n}}\right)\tag{*}$$from
$$
I(n+1)=\frac{1}{\sqrt{2}} n^{n/2}e^{-n/2+\...
- 3,047
0
votes
0
answers
24
views
Asymptotic Behaviour of $e^{ \alpha \log{( \frac{a}{b})} \log(n)} \times \sum_{t=1}^{n/2} e^ {- \frac{2t}{3} (\log(t) - 3) }$
If $a>b>0$ and $\alpha = \frac{x+1}{2y}$ where $y>0$ and $x\geq 0$. Under which conditions the following summation is asymptotically ($n\to \infty$) upper-bounded by $ k n$ where $k$ is a ...
- 21
2
votes
1
answer
25
views
Bounding $f'(z)$ with $O(\log(\frac{1}{1-r}))$ for an Analytic Series
I am working with an analytic function defined within the unit disk $|z| < 1$ as follows:
$$
f(z) = \sum_{n=1}^{\infty} a_{n} z^{n},
$$
where I have the condition that $\sum_{n=1}^{N} n|a_{n}| = O(\...
- 81
3
votes
1
answer
69
views
Behaviour of polylogarithm at $|z|=1$
I have the sum
$$
\sum_{n=1}^\infty \dfrac{\cos (n \theta)}{n^5} = \dfrac{\text{Li}_5 (e^{i\theta}) + \text{Li}_5 (e^{-i\theta})}{2},
$$
where $0\leq\theta < 2 \pi$ is an angle and $\text{Li}_5(z)$ ...
