Tagged Questions

-2
votes
0answers
40 views

Solve $ z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots $.

I am trying to solve $z\in \mathbb{C}$ in terms of $a\in \mathbb{C}$, where $$ z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots. $$ I plugged $z= \sum_{k=0}^\infty c_k a^k $ into the ...
6
votes
4answers
111 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
224 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
45 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
20 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 ...
17
votes
6answers
937 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
16 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 ...
4
votes
2answers
259 views

Showing that $\lim_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$

How to show that $$\lim_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$$ No clue at all. Need help! Appreciated!
11
votes
3answers
249 views

Closed form for $\sum_{k=0}^{n} k\binom{n}{k}\log\binom{n}{k}$

Is it possible to write this in closed form: $$\sum_{k=0}^{n} k\binom{n}{k}\log\binom{n}{k}$$ Can you get something like $$n2^{n-1}\log(2^{n-1})$$
5
votes
1answer
235 views

Missing term in series expansion

I asked a similar question before, but now I can formulate it more concretely. I am trying to perform an expansion of the function $$f(x) = \sum_{n=1}^{\infty} \frac{K_2(nx)}{n^2 x^2},$$ for $x \ll ...
2
votes
3answers
57 views

Solve using the definition of Big-$\mathcal{O}$

By using the definition of Big-$\mathcal{O}$, show that \begin{align} 2^{(n+2)} + 3^{(n+1)} & \text{ is } \mathcal{O}(3^n)\\ \sqrt{10n^2+7n+3} & \text{ is } \mathcal{O}(n)\end{align} I'm ...
0
votes
0answers
45 views

How can weakly/strongly decreasing or increasing approximate sums be explained?

I'm reading around big O to get some concept about performance for data structures. The mathematics book recommended by the open book ~ maths for computer science In the book (pg 456) part of the ...
1
vote
1answer
60 views

Asymptotic behaviour of a sequence

We fix $\alpha >0$, and we look for the asymptotic behaviour when $n \to +\infty$ of $$u_n=1^{\alpha n}+2^{\alpha n}+\cdots+n^{\alpha n}.$$ Any suggestion?
3
votes
4answers
144 views

Simple proof of showing the Harmonic number $H_n = \Theta (\log n)$

Consider the partial sum of the divergent Harmonic series $H_n = \sum\limits_{k = 1}^{n}\frac{1}{k}$. I recently saw a question which required finding out the asymptotic bounds of $H_n$. Now, I could ...
3
votes
6answers
183 views

Asymptotic expansion of $ I_n = \int_0^{\pi/4} \tan(x)^n \mathrm dx $

I'm trying to compute the asymptotic expansion of $$ I_n = \int_0^{\pi/4} \tan(x)^n \mathrm dx $$ Here is what I've done: Change of variable $$ t= \tan x $$ $$ I_n = \int_0^1 \frac{t^n \mathrm ...
2
votes
1answer
55 views

Asymptotic expansion of $ u_n = \int_0^1 \ln(1+t^n) \mathrm dt $

I would like to know how I can compute the asymptotic expansion of: $$ u_n = \int_0^1 \ln(1+t^n) \mathrm dt $$ Using the dominated convergence theorem, we get: $$ u_n \sim \frac{\pi^2}{12n}$$ How ...
3
votes
2answers
66 views

Asymptotic rate of growth of a sum

Consider $$\Phi_0(x) = \sum_{i=0}^{\infty} (1-x)^i,$$ where $x \in (0,1)$. As $x \rightarrow 0$, $\Phi_0(x)$ blows up as $\Theta(1/x)$. Similarly, consider $$ \Phi_1(x) = \sum_{i=0}^{\infty} i ...
3
votes
4answers
121 views

Studying $ u_n = \int_0^1 (\arctan x)^n \mathrm dx$

I would like to find an equivalent of: $$ u_n = \int_0^1 (\arctan x)^n \mathrm dx$$ which might be: $$ u_n \sim \frac{\pi}{2n} \left(\frac{\pi}{4} \right)^n$$ $$ 0\le u_n\le \left( \frac{\pi}{4} ...
11
votes
1answer
146 views

If $\lambda_n \sim \mu_n$, is it true that $\sum \exp(-\lambda_n x) \sim \sum \exp(-\mu_n x)$ as $x \to 0$?

If $\lambda_n,\mu_n \in \mathbb{R}$, $\lambda_n \sim \mu_n$ as $n \to +\infty$, and $\mu_n \to +\infty$ as $n \to +\infty$, is it true that $$ \sum_{n=1}^{\infty} \exp(-\lambda_n x) \sim ...
4
votes
2answers
264 views

Asymptotics of system of linear equations

I have a system of linear equations as follows. $$M(p) = 1+\frac{n-p-1}{n}M(n-1) + \frac{2}{n} N(p-1) + \frac{p-1}{n}M(p-1)$$ $$N(p) = 1+\frac{n-p-1}{n}M(n-1) + \frac{p}{n}N(p-1)$$ $$M(1) = ...
7
votes
3answers
148 views

Asymptotic expansion of a series

I am interested in the asymptotics, as $x$ tends to $0$, of $$f(x) = \sum_{n=1}^\infty \frac{1}{n}\frac{1}{(e^{nx}-1)^2}$$ This function is well defined for every $x > 0$ (for example, use ...
2
votes
1answer
45 views

Series expansion and approximate solution

I have the following equation: $a(x)k-b(x)=0\Rightarrow k=\dfrac{b(x)}{a(x)}$. I find approximate solutions around $x=0$ by two ways: (1) I expand the equation as ...
16
votes
2answers
705 views

please solve a 2013 th derivative question?

$ f(x) = 6x^7\sin^2(x^{1000}) e^{x^2} $ Find $ f^{(2013)}(0) $ A math forum friend suggest me to use big O symbol, however have no idea what that is, so how does that helping?
2
votes
2answers
161 views

limit of $\frac{(2n)!}{4^n(n!)^2}$

I'd love to understand the behaviour of the sequence $$ \frac{(2n)!}{4^n(n!)^2} \text{as } n \to \infty $$ the first step would be to simplify this to $$ \frac{(2n)(2n-1)(2n-2)\cdots(n+1)}{4^n \cdot ...
2
votes
2answers
49 views

Study of a series of functions

I've to study this series: $$\sum_{n=1}^\infty e^{\sqrt n\,x}$$ My teacher wrote that with the asymptotic comparison with this series: $$\sum_{n=1}^\infty\frac{1}{n^2}$$ My series converges ...
2
votes
4answers
196 views

Asymptotic behavior of $(1/2 + 2/3 + 3/4 + 4/5 + \cdots + (n-1)/n ) \times n$

I am interested in the following questions: given: $$G(n) = \left(\frac12 + \frac23 + \frac34 + \frac45 + \cdots + \frac{n-1}n\right)n$$ what is a $F(n)$ which could be an upper bound (clearly ...
2
votes
1answer
112 views

Asymptotics of $\sum\limits_{j_1,\ldots, j_{2k}\neq n}\frac{1}{(n-j_1)(n-j_2)\cdots(n-j_{2k-1})(n-j_{2k})}$

How to estimate the following sum in terms of $n$? $$ \sum_{j_1,\ldots, j_{2k}\neq n}\frac{1}{(n-j_1)(n-j_2)\cdots(n-j_{2k-1})(n-j_{2k})}$$ with $n+j_1, j_1-j_2, \ldots, j_{2k}-j_{2k-1}, n-j_{2k} \in ...
2
votes
3answers
122 views

Studying $ u_{n}=\frac{1}{n!}\int_0^1 (\arcsin x)^n \mathrm dx $

I would like to find a simple equivalent of: $$ u_{n}=\frac{1}{n!}\int_0^1 (\arcsin x)^n \mathrm dx $$ We have: $$ 0\leq u_{n}\leq \frac{1}{n!}\left(\frac{\pi}{2}\right)^n \rightarrow0$$ So $$ ...
4
votes
2answers
78 views

Showing that $ \sum_{k=n+1}^{\infty} e^{-k^2} \sim e^{-(n+1)^2}$

I would like to show that: $$ \sum_{k=n+1}^{\infty} e^{-k^2} \sim e^{-(n+1)^2}$$ We have: $$ \forall p\geq2$$ $$ \exp((n+1)^2-(n+p)^2)=\exp(2n(1-p)+1-p^2)=o(1)$$ $$ e^{-(n+p)^2}=o \left( ...
0
votes
2answers
46 views

Find the rate of growth for $\sum_{n=1}^N 1/n^p$ in term of big $O$ notation

Find the rate of growth for $$ \sum_{n=1}^N \frac{1}{n^p} $$ in term of big $O$ notation for the three cases $0 < p < 1$, $p=1$ and $p>1$. It seems the question can be approached by ...
4
votes
2answers
84 views

Asymptotic growth of $\sum_{i=1}^n \frac{1}{i^\alpha}$?

Let $0 < \alpha < 1$. Can somebody please explain why $$\sum_{i=1}^n \frac{1}{i^\alpha} \sim n^{1-\alpha}$$ holds?
3
votes
1answer
70 views

Estimate a sum of products

Let $0 < \alpha < 1$. Show that for $\lambda > 0$ big enough $$\sum_{n=1}^\infty \prod_{k=1}^n \frac{1}{\lambda k^{-\alpha} + 1} < \infty$$ I think $\lambda = 1$ is enough. You could ...
1
vote
1answer
71 views

Big O with Log base equivalences and a question about sum of series.

Hi I need help figuring these out: True or False: $\log_2 n$ is $O(\log_3 n)$ I used the definition of Big O in Dasgupta's book: ${f(n)}\over g{(n)}$ $\leq c$ So I used the base transformation rule ...
5
votes
3answers
138 views

Is there a minimal diverging series?

Is there a function $f:\mathbb{N} \to \mathbb{R}^+$ s.t. its series $\Sigma_{i=0}^\infty f(n)$ diverges but the series for all function in $o(f)$ converge?
0
votes
2answers
51 views

Is there a diverging series in $o(1/n)$

Is there a function $f : \mathbb{N} \to \mathbb{R}^+$ in $o(1/n)$ s.t. $\Sigma_{i=0}^\infty f(n)$ diverges?
2
votes
1answer
76 views

What is the value of this summation in Big O terms?

I am trying to do an analysis for the cost of n inserts into a hashtable datastructure and I have a factor like the one below: $$\sum_{i=0}^{\lfloor\lg {(n-1)}\rfloor} 2^i$$ What will be the Big O ...
0
votes
1answer
48 views

Asymptotic behavior of a sequence based on a subsequence II

Let $\{a_{n}\}$ be a non-increasing sequence of positive numbers. if for some positive integers $l,p$ and $R>1$ we have $a_{(ln)^{p}}=O(R^{-n})$ as $n\to\infty$, what can we say about the behavior ...
3
votes
3answers
271 views

Predicting the next vector given a known sequence

I have a sequence of unit vectors $\vec{v}_0,\vec{v}_1,\ldots,\vec{v}_k,\ldots$ with the following property: $\lim_{i\rightarrow\infty}\vec{v}_{i} = \vec{\alpha}$, i.e. the sequence converges to a ...
2
votes
1answer
92 views

Monotone Sequence & Big-O Notation

I was thinking about the following problem: Consider a sequence $a_n \to 0, a_n >0$ which is monotone, i. e. $a_n\ge a_{n+1}$. Now suppose for every $C>0$ there is a subsequence $n(k)$ such ...
2
votes
1answer
63 views

Asymptotic behavior of a sequence based on a subsequence.

Let $c\in(0,1)$, $m\geq 1$ be positive integer and $\{a_{n}\}$ a decreasing sequence of positive real numbers. Suppose that $$a_{n^{m}}\leq K c^{n}n^{-m/2}, \forall n\in\mathbb{N}, $$for some ...
0
votes
1answer
62 views

Decay for the tail of a series.

Let $p>1$. I would like to have an estimate for the decay of the sequence $s_{n}=\sum_{k=n}^{\infty}k^{-p}$. Does anyone know of a bound of this type in the literature? Thanks!
1
vote
2answers
72 views

Computing limits with Asymptotics (Book suggestion)

Is there a standard book or reference to learn techniques of computing limits similar to the answer of this problem: How does one easily compute the limit of $a_n=(n\cdot \ln(\frac{n+1}{n}))^n$? ...
4
votes
3answers
118 views

Equivalent of $ I_{n}=\int_0^1 \frac{x^n \ln x}{x-1}\mathrm dx, n\rightarrow \infty$

I would like to show that $$ I_{n}=\int_0^1 \frac{x^n \ln x}{x-1}\mathrm dx \sim_{n\rightarrow \infty} \frac{1}{n}$$ Using the change of variable $u=x^n$: $$ I_{n}=\frac{1}{n^2} \int_0^1 ...
3
votes
1answer
153 views

Series about Euler-Maclaurin formula

The Euler-Maclaurin formula says (from Concrete Mathematics section 9.5) \[ \sum_{a\le{}k< b}f(k)=\int_a^bf(x)dx+\left.\sum_{k=1}^m\frac{B_k}{k!}f^{(k-1)}(x)\right|_a^b+R_m \] where ...
2
votes
3answers
132 views

Equivalent of $ u_{n}=\sum_{k=1}^n (-1)^k\sqrt{k}$

I'm trying to show that $$ u_{n}=\sum_{k=1}^n (-1)^k\sqrt{k}\sim_{n\rightarrow \infty} (-1)^n\frac{\sqrt{n}}{2}$$ when $n\rightarrow\infty$ How can I first show that $$u_{2n}\sim_{n\rightarrow ...
7
votes
2answers
139 views

Equivalent of $\int_0^{\infty} \frac{\mathrm dx}{(1+x^3)^n},n\rightarrow\infty$

According to my calculations $$ \int_0^\infty \frac{\mathrm dx}{(1+x^3)^n}=\frac{(3n-4)\times(3n-7)\times\cdots\times5\times2}{3^{n+1/2}(n-1)!}2\pi$$ How can an equivalent of $$ \int_0^\infty ...
0
votes
1answer
88 views

Equivalent of $ u_{n}=\int_0^{\pi/2} \cos\left(\frac{\pi}{2}\sin(x)\right)^n \mathrm dx $

I would like to find an equivalent of the sequence $u_{n}$ where $$ u_{n}=\int_0^{\pi/2} \cos\left(\frac{\pi}{2}\sin(x)\right)^n \mathrm dx $$ The substitution $x\rightarrow \frac{\pi}{2}\sin(x)$ ...
5
votes
1answer
195 views

expansion of $\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt$ in inverse powers of $p$

This question relates to this answer I gave to a question about the integral $$\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt\;.$$ I derived an expansion in inverse powers of $p$ and then ...
3
votes
0answers
94 views

Asymptotics of $\sum_{n=1}^{\infty}x^{n+1/n}/n!>e^x$?

Is there a way to find precise asymptotics or better bounds of series such as $\sum_{n=1}^{\infty}x^{n+1/n}/n!>e^x$ ? Or $\sum_{n=1}^{\infty}x^{\sqrt n}/e^n$?
2
votes
1answer
57 views

Finding an equivalent of $ u_{n}=\prod_{k=1}^{n} k^k $

I would like to find an equivalent of: $$ u_{n}=\prod_{k=1}^{n} k^k $$ I managed to find and asymptotic expansion of $ \ln(u_{n}) $ whose precision is $ o(n) $: $$ ...

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