8
votes
1answer
95 views

How prove this sequence $a_{n}=\sqrt{n}+\frac{1}{2}-\frac{1}{8\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right)?$

let sequence $\{a_{n}\}$ such $$a_{1}=1,a_{n+1}=1+\dfrac{n}{a_{n}}$$ show that: $$a_{n}=\sqrt{n}+\dfrac{1}{2}-\dfrac{1}{8\sqrt{n}}+o\left(\dfrac{1}{\sqrt{n}}\right)?$$ This result is china student ...
2
votes
1answer
37 views

asymptotics of this sum $ x \to 0 $

given the sum $$ \sum_{n=0}^\infty \frac{\exp(-nx)}{n+a} =f(x) $$ what would be the asymtptic of this series ?? for $a=1$ i believe this series goes as $ f(x) \sim \frac{1}{x}+ \gamma $ for every ...
0
votes
1answer
24 views

Is it possible to integrate this asymptotic expansion?

Suppose that some smooth function $f \in C^\infty\bigl(\mathbb R^n \times (0,+\infty)\bigr)$ possesses an asymptotic development $$ f(x,t) \sim t^{-\alpha} e^{ith(x)} \sum\limits_{k=0}^{+\infty} ...
4
votes
0answers
55 views

Asymptotic expansion of $\zeta(s)$

It is well known that $$ \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
2
votes
0answers
267 views

First disagreement in PROUHET THUE MORSE exponentially big?

Let two sequences of integers be $a_1, \cdots, a_n$ and $b_1, \cdots, b_n$ such that with $a_i \in \{1, \cdots n\}$ and $b_i \in \{1, \cdots, n\}$. Let $k$ be the min integer such that $\sum_{i=1}^n ...
0
votes
1answer
43 views

Show that $3^n = 2^{O(n)}$ [duplicate]

The formal description I have is that this is this: $f(n) = n^{O(n)}$ iff there exists some $h(n) = O(n)$ such that $f(n) = n^{h(n)}$. I don't see how this can be applied to the problem to show that ...
5
votes
1answer
50 views

Asymptotics of A030283

I wondered about the following sequence $a_i, i \in \mathbb N$ today: $a_1=1$ $a_n={\text{Smallest integer} > a_{n-1} \text{ that does not share any decimal digits with } a_{n-1}}$ The first ...
3
votes
0answers
49 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
1
vote
3answers
113 views

Equation with the big O notation

How I can prove equality below? $$ \frac{1}{1 + O(n^{-1})} = 1 + O({n^{-1}}), $$ where $n \in \mathbb{N}$ and we are considering situation when $n \to \infty$. It is clearly that it is true. But I ...
1
vote
1answer
84 views

Mysterious subleading corrections to sums with internal dependence on limit

Is there a standard method for finding expansions in $N$ of sums like $$S(N)=\sum_{n=0}^N \sqrt{N^2-n^2}$$ beyond the first term? It is easy to compute here that $$S(N)=N^2 \int_0 ^1 \sqrt{1-x^2} ...
1
vote
1answer
20 views

Proving bigO of the series $c^k$ as $c$ goes to $k$

The same question was posted here but I feel necessary details were left out: Geometric series and big theta The question states: Show that, if $c$ is a positive real number, then $f(n)= ...
4
votes
1answer
110 views

Recursive sequence with binomial coefficients

I have a sequence $\epsilon_i$ defined recursively for $i\ge 1$ as follows \begin{eqnarray*} \epsilon_1 &=& \frac{1}{p}\\ \epsilon_n &=& \frac{1}{1-(1-p)^n}\left( 1 + \sum_{j=1}^{n-1} ...
4
votes
1answer
29 views

Find asymptotics for solution $x$ of $(x+1)^{\frac{n+1}{n}}-x^{\frac{n+1}{n}}=5$

It is easy to see that for any $n\geq 1$, the equation $(x+1)^{\frac{n+1}{n}}-x^{\frac{n+1}{n}}=5$ has a unique positive solution ; call it $x_n$. Is there a simple asymptotic formula for $x_n$ ? I ...
3
votes
1answer
50 views

The statements $f(n) = O(n^{\epsilon})$ for all $\epsilon > 0$ and $f(n) = n^{o(1)}$.

Consider the statements \begin{align} \tag{A} f(n) &= O(n^{\epsilon}) \text{ for all } \epsilon > 0 \\ \tag{B} f(n) &= n^{o(1)} \end{align} Questions: It's clear that (B) implies (A). ...
3
votes
1answer
53 views

Equivalent of a recurrence sequence [duplicate]

Let $x_{0} = 2$ and $x_{n+1} = x_{n} + \ln(x_{n})$, how can I find an asymptotic equivalent of this sequence say, to the third term? (This is not homework, it was a problem in the Oral Examination ...
4
votes
0answers
144 views

Double harmonic summation

I am interested in determining an asymptotic formula for the double summation of $1/(ab)$, where $a$ is an odd integer ranging between 1 and $k/\sqrt{j}$, $b$ is an odd integer ranging between $a$ and ...
3
votes
3answers
187 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
8
votes
3answers
103 views

Sum $S(n,c) = \sum_{i=1}^{n-1}\dfrac{i}{ci+(n-i)}$

Consider the sum $$S(n,c) = \sum_{i=1}^{n-1}\dfrac{i}{ci+(n-i)}$$ where $0\le c\le 1$. When $c=0$, $S(n,c)$ grows asymptotically as $n\log n$. When $c=1$, $S(n,c)$ grows asymptotically as $n$. ...
0
votes
1answer
30 views

Determine convergence or divergence of a slowly increasing sequence

We have by the integral test: $\sum_{n=2}^{\infty}\frac{1}{n}=\infty$ $\sum_{n=2}^{\infty}\frac{1}{n\ln n }=\infty$ $\sum_{n=2}^{\infty}\frac{1}{n\ln n (\ln\ln n) }=\infty$ ...
6
votes
1answer
90 views

Limits of $\sum_{m=1}^{+\infty} \sum_{n=1}^{+\infty}e^{-mn x}$ at $0$ and $\infty$

Let $f(x) = \sum_{m=1}^{+\infty} \sum_{n=1}^{+\infty}e^{-mn x}$ for $x > 0$. Prove that $f(x) \sim e^{-x}$ as $x \to \infty$ and $\lim_{x\to 0} x\cdot (f(x) + \frac{1}{x}\log x)= \gamma$ where ...
2
votes
1answer
90 views

How to evaluate $\sum\limits_{k=0}^{n} \sqrt{\binom{n}{k}} $

Can we find $$ \sum_{k=0}^{n} \sqrt{\binom{n}{k}} \quad$$ This problem asked me my friend about a year ago, but I didn't know how to attack problem. Now, I am interesting in solution. Any suggestion? ...
1
vote
1answer
42 views

How to prove that sum given by generating function diverges for given value of $x$

I have a generating function: $A(x)=\dfrac{3-8x}{1-4x+6x^2-3x^3}$ (also I have a recurrence from which this function is built). I have to prove that sum $\sum\limits_k a_k\left(\dfrac{4}{3}\right)^k$ ...
3
votes
2answers
139 views

Show $S(t) =\sum_{n=-\infty}^\infty\sin{(n^2t^2)}e^{-tn^2}$ is $O(t^p)$ at zero

An old qualifying exam problem: For $t>0$, define $$S(t) =\sum_{n=-\infty}^\infty\sin{(n^2t^2)}e^{-tn^2}.$$ Show that $S(t) = C t^p + o(t^p)$ as $t\to 0$ . Find $C$ and $p$. There are a couple of ...
2
votes
2answers
21 views

Growth of series with decreasing numerators and increasing deonimators

It is known that $$H(n)=1+\dfrac12+\ldots+\dfrac1n$$ grows with the same rate as $\log n$. Therefore, $$nH(n)=n\left(1+\dfrac12+\ldots+\dfrac1n\right)=\frac n1+\frac n2+\ldots+\frac nn$$ grows with ...
3
votes
1answer
185 views

Asymptotic approximation of sum $\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$

I need to find the asymptotic approximation of this sum $$\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$$ Can you please share a link to theory or hint how it can be solved? Here is my attempt $n ...
3
votes
1answer
68 views

Asymptotics of sequence depending on Tricomi's function

I'm dealing with the following sequence $$ p_n = U(a, a - n, 1)$$ where $a > 0$ and $U$ is Tricomi's function. I suspect that asymptotically when $n \to \infty$ its behaviour is a power law ...
1
vote
0answers
76 views

Questions about the superfactorial function.

N superfactorial or $n\$$ is defined as - $$n\$=\prod_{k=1}^n k!$$ Then is there any asymptotic formula for this? Are there any infinite series , integrals related to this function? Is there a ...
0
votes
0answers
22 views

using series and big o notation [duplicate]

Recall the equivalence: $$m = 2^k , k = \log_2 m$$ (a) Consider the sequence: $$a_1 = 1; a_{k+1} = 2a_k$$ What is the smallest $k$ for which $a_k \geq n$? Your answer should be a function of $n$, and ...
0
votes
1answer
386 views

Geometric series and big theta

Consider the following function: $$S(n)=1+ c + c^2 + ··· + c^n,$$ where c is a positive real number. (A) This function is the sum of a geometric series. Give a precise closed-form formula for ...
9
votes
1answer
92 views

Find asymptotic of recurrence sequence

Given a sequence $x_1=\frac{1}{2}$, $x_{n+1}=x_n-x_n^2$. It's easy to see that it limits to $0$. The question is: is there exists an $\alpha$ such, that $\lim\limits_{n\to\infty}n^\alpha x_n\neq0$. ...
2
votes
0answers
71 views

Upper Bound on $\frac{1}{1-\beta u}-\sum\limits_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$

Is there any procedure to find an upper bound of the following expression? $$\frac{1}{1-\beta u}-\sum_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$$ Here $u,\beta\in\mathbb{R},\ u>1,\ ...
2
votes
0answers
52 views

Bounds on a rapidly increasing sequence

I read about a sequence similar to this one here on Stack Exchange a while back, somebody used it as an example for something that I can't recall! However, when I read about it it made me come up with ...
3
votes
2answers
136 views

On proving the convergence of $1/n^2\sum_{1\le k\le n}\varphi(k)$

Let $$\Phi_n=\frac{1}{n^2}\sum_{k=1}^n\varphi(k).$$ How one can show that $\Phi_n$ is convergent sequence? (Here, $\varphi$ denotes the Euler's totient function.) And please, without any monster ...
4
votes
1answer
60 views

Is there a “natural” subsequence of positive integers $k_1 < k_2 < \ldots$ such that $\sum_{i=1}^n \frac{1}{k_i} = \Theta (\log \log \log n)$?

The harmonic series partial sums grow like $\log n$, and the sum of inverses of the first $n$ primes grows like $\log \log n$. Is there an example of a "nautral" subset of the positive integers (say ...
7
votes
5answers
388 views

The asymptotic expansion for the weighted sum of divisors $\sum_{n\leq x} \frac{d(n)}{n}$

I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum_{d|n}$ is the number of divisors of the integer $n$. It is well known that ...
23
votes
1answer
437 views

Power towers: to infinity and all the way back

In the following, let $n$ be a positive integer, all other variables be real (furthermore, $a>1$), all functions be real-valued, and logarithms of negative arguments be undefined. Let ...
2
votes
1answer
51 views

Growth rate of Taylor convergents near pole

For any fixed $z_0\in\mathbb{C}\setminus \{0\}$ and $\beta\in\mathbb{R}^{+}$, prove that $$\left.T_n\left(\log^{\beta}z;z_0\right)\right|_{z=0}\sim\log^{\beta} n$$ Note: I observed that this holds ...
3
votes
2answers
70 views

Prove: For all $n\geq 1$, $a_{n+1}-a_n<8^ka_n^{\left(1-\frac{1}{k}\right)^3}$.

Set $S=\left\{\left.x^k+y^k+z^k\right|x,y,z\in Z^+\cup \{0\}\right\}$, k is a positive integer, sort elements of $S$ increasingly, that $a_1<a_2<a_3<\text{...}<a_n<\text{...}$. Prove: ...
0
votes
1answer
84 views

infinite sum with each summand converging to zero almost surely

Suppose $X_n$ is a random variable such that $X_n=O(b_n)$ almost surely, with $b_n\to 0$ as $n\to \infty$. Let $C$ be a real constant and $S_{j,k}(X_n)=\sum_{i=j}^kC^iX_n^i$ for ...
2
votes
2answers
199 views

$\sum _{p\leq n}\frac{1}{p}=C+\ln \ln n+O\left(\frac{1}{\ln n}\right)$

$\sum _{p\leq n}\frac{\ln p}{p}=\ln n+O(1),n\geq 2,$ where $p$ is a prime number, prove: $$\sum _{p\leq n}\frac{1}{p}=C+\ln \ln n+O\left(\frac{1}{\ln n}\right)~~~(1)$$ one examination ...
2
votes
0answers
55 views

Asymptotics and Related Properties

I have a rather general question: If there are two integer sequences such that $$\lim_{n\to\infty}A(n)/n=\lim_{n\to\infty}B(n)/n=c$$ is there anything else that can be said about them necessarily? ...
9
votes
3answers
247 views

Closed form of $\sum\limits_{i=1}^n k^{1/i}$ or asymptotic equivalent when $n\to\infty$

Is there a "closed form" for $\displaystyle S_n=\sum_{i=1}^n k^{1/i}$ ? (I don't think so) If not, can we find a function that is asymptotically equivalent to $S_n$ as $n\to\infty$ ?
3
votes
1answer
172 views

Asymptotics of a summation over real valued functions

Let $f$ and $g$ be integrable in $[0,1]$ and $(-\infty, \infty)$ respectively. Let $a_k$ be a divergent series of positive terms and $S_k = a_1 + a_2 + \ldots + a_k$ such that the following ...
1
vote
4answers
164 views

How is “n+n/2+n/4…1” equal to “2n-1” using the formula for geometric series?

I never knew not having good knowledge of basic maths will be so crippling!! So please help me out this time. I'll be working on my maths from today on. I was discussing about complexity of an ...
7
votes
4answers
257 views

Sum of kth roots ($\sum\sqrt[k]{m}$)

I'm trying to find an asymptotic to $$S(n) = \sum_{k=1}^n\sqrt[k]{m}$$ From computational tests, it seems to grow nearly as slowly as $n$. However even $$\sum_{k=1}^\infty\sqrt[k]{m}-1$$ diverges (for ...
9
votes
0answers
314 views

Asymptotic related to the infinite product of sine

The amount is somewhat complicated ($x$ is a constant): $$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$ I want to enrich my handy powerful ...
2
votes
2answers
49 views

Prove the following: Product of Roots

$1^{(1/1)} \cdot 2^{(1/2)} \cdot 3^{(1/3)} \cdot 4^{(1/4)} \cdot 5^{(1/5)} $.... diverges well I don't really know if it does but my gut tells me it does: I can take the log of this product to ...
1
vote
1answer
51 views

How do we determine as to how long we should sum an asymptotic series of a function to get the answer correct up to a particular precision?

As an example, consider the asymptotic expansion for polygamma function . What should be the min value of 'k' in the equation to get the answer correct upto a particular precision, say pth. Is there ...
18
votes
6answers
1k views

Is there a formula for $\sum_{n=1}^{k} \frac1{n^3}$?

I am searching for the value of $$\sum_{n=k+1}^{\infty} \frac1{n^3} \stackrel{?}{=} \sum_{n = 1}^{\infty} \frac1{n^3} - \sum_{n=1}^{k} \frac1{n^3} = \zeta(3) - \sum_{n=1}^{k} \frac1{n^3}$$ For which ...
1
vote
1answer
29 views

Questions on Strong mixing coefficients that satisfy $\alpha(m) = O(m^{-a-\epsilon})$

Say that we have strong mixing coefficients that satisfy the following: $\alpha(m) = O(m^{-a-\epsilon})$ for some $\epsilon > 0$. If we have $h\in {\mathbb N}$ that is finite and $h>m$, I have ...