All Questions
28
questions
3
votes
1
answer
85
views
Proving two averages are asymptotically equivalent
Suppose $f(n)\sim g(n)$ as $n\to\infty$. Is it necessarily true that \begin{equation}\frac{1}{n}\sum_{k=1}^n|f(k+1)-f(k)|\sim\frac{1}{n}\sum_{k=1}^n|g(k+1)-g(k)|\end{equation}
as $n\to\infty$?
...
- 31
1
vote
1
answer
79
views
Asymptotic equivalent of $\sum_{k=1}^n a^k k^{-1/2}$
I encountered recently the following partial sum $\sum_{k=1}^n a^k k^{-1/2}$ with $a$ a constant approximately equal to $2.955$.
I was wondering if there were any clever way to find an asymptotic to ...
- 762
1
vote
0
answers
44
views
Showing that the large $x$ limit of $e^{-x}$ is zero by considering the negative gradient of the partial sums
I know that
$$\lim_{x \to \infty} e^{-x}=0$$
If I consider the partial sums of the respective Maclaurin series,
$$
\sum_{i=0}^{n}\frac{(-1)^{i} x^i}{i!}, \,\, n=1,2,\dots
$$
its seems that despite ...
- 2,797
3
votes
1
answer
75
views
Asymptotic formula of $\sum_{r=1}^{n}\frac{(-1)^{r-1}}{(r-1)!}{n\choose r}$
I need to find the asymptotic equivalence of the sum $$\sum_{r=1}^{n}\frac{(-1)^{r-1}}{(r-1)!}
{n\choose r} $$ where ${n\choose r}$ is the binomial coefficient.
We have the binomial identity $$(1-x)^...
user1094088
1
vote
0
answers
54
views
Asymptotic development of a sum [duplicate]
I'd like to find an asymptotic development for
$\sum_{k=2}^{n} \ln(k)$
I know this sum is equivalent to $n\ln(n)-n$ but I cannot find more terms.
3
votes
0
answers
129
views
The limit of an oscillating sum of powers of $x$
I am trying to evaluate the following limit of an oscillating sum,
$$
S=\lim_{x \to \infty}\sum _{i=3}^{\infty} \frac{(-1)^{i-1} x ^{i-\frac{4 (i-1)}{i+1}}}{(i-1)!}
$$
which looks like
$$
\lim_{x \to \...
- 2,797
3
votes
3
answers
135
views
Upper bound of the sum $\sum_{n=1}^\infty \tan^{-1}\left(\frac{x}{1-(x-n)^2}\right)$ as $x\to \infty$
I am stuck at finding a Big-O upper bound of the following sum as $x\to \infty$ $$\sum_{n=1}^\infty \tan^{-1}\left(\frac{x}{1-(x-n)^2}\right)$$
I tried: Since $$ \sum_{n=1}^\infty\tan^{-1}\left(\frac{...
user1082747
5
votes
2
answers
360
views
Asymptotic analysis of $\sum_{k=2}^n\frac{k}{\ln k}$
I know that by integrating, we can show $$\sum_{k=1}^n \ln(k)\sim n\ln n-n$$
Now what can we do for $$\sum_{k=2}^n\frac{k}{\ln k}$$ which doesn't admit an integral approximation in analytic form?
(Or ...
- 910
4
votes
0
answers
97
views
Convergence rate to the average value for a sequence that is uniformly distributed modulo $1$
Let $x_n$ be a sequence of real numbers that asymptotically follow a uniform distribution modulo $1$ (for example, $x_n=n\sqrt{2}$ with $n$ positive integer $\leq N$ and $N\rightarrow \infty$). It is ...
- 17.1k
2
votes
1
answer
161
views
Asymptotic formula for a sum involving the exponential function.
Let $n$ be a positive integer, and let $c$ be a positive constant. Consider the asymptotic estimate
$$
\sum_{1\le kv < n}\frac{ve^{c\sqrt{n-kv}}}{n-kv} = e^{c\sqrt{n}}\left(1 + \mathcal{O}\!\left(\...
- 3,296
1
vote
0
answers
20
views
Justifying Changes of Variables in Closed Forms of Partial Sums of Non-Negative Sequences of Real Numbers
For any integer $n\geq1$, let $\#_{1}\left(n\right)$ denote the number of $1$s digits in the binary representation of $n$. It is easy to show that the following generating function identity holds:$$\...
- 2,219
1
vote
3
answers
154
views
Why does the Big O notation of summation give log
I’m not understanding the following line in a proof
I believe it should be $O(h^2\psi(2h))$ since $\sum_{j=1}^{h}\frac{1}{j}\leqslant h\cdot1=h$
- 1,280
1
vote
1
answer
40
views
A question on asymptotic analysis
Consider the function $f:\mathbb{R}^+\times\mathbb{N}\to\mathbb{R}$. It is known that, for any $\lambda\in\mathbb{R}^+$ and $\eta\in\mathbb{N}$, $$f(\lambda,\eta) \ge 0 $$
Also $$\lim\limits_{\lambda\...
- 4,247
1
vote
0
answers
65
views
Asymptotic growth bound on a sequence equivalent to an asymptotic growth bound on its sum.
Let $(a_n)$ and $(b_n)$ be sequences of positive real numbers. Denote by $o$ the "little-oh" Landau symbol. Is it possible, in general, to find a third sequence $(c_n)$ such that $\sum_{k=0}^n a_k = o(...
- 4,795
1
vote
3
answers
101
views
The following seems true for a wide range of functions: $\sum_{k=1}^n f(k)f(n-k) = n f^2(n/2) (1 + o(1))$
I haven't worked out all the details yet, but it seems to be true for the following functions:
$f(k) = 1$
$f(k) = 1/k!$
$f(k) = a^k$
$f(k) = 1/\log(k+1)$
What are the conditions on $f$ for this to ...
- 3,645