3
votes
0answers
67 views

Closed form $\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}}$ and $\sum_{n=2}^{\infty} \frac{n}{\ln^n{n}}$

Apologies if this has been asked before, but I was playing around with Wolfram Alpha and got approximations but not closed forms for $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109 $$ ...
2
votes
1answer
29 views

Does $\sum_{i=1}^{k-1}\lceil \log_2\frac{N}{i}\rceil$ have a closed form?

Does the following have a closed formula? $$\sum_{i=1}^{k-1}\left\lceil \log_2\frac{N}{i}\right\rceil$$
1
vote
1answer
45 views

$\sum_{x=a}^{b-1}\frac{1}{x}$ and $\sum_{x=a+1}^b\frac{1}{x}$

I have to prove the following relations: $\sum_{x=a}^{b-1}\frac{1}{x}\geq\log b - \log a $ $\sum_{x=a+1}^{b}\frac{1}{x}\leq\log b - \log a $ I tried to use the relation that $\int_a^b \frac{1}{x} ...
2
votes
4answers
456 views

Infinite sum of logarithms

Is there any closed form for this expression $$ \sum_{n=0}^\infty\ln(n+x) $$
0
votes
0answers
37 views

Are generating functions ever analytic for logarithmic series?

Given a series $s_n = \ln(n) f(n)$ where $f(\cdot)$ is an elementary analytic function which does not involve the logarithm. More precisely $f$ can have simple poles but no branch cuts or essential ...
2
votes
3answers
97 views

Summation with Ceilinged Logarithmic Function

According to Johann Blieberger's paper - "Discrete Loops and Worst Case Performance" (1994): $$ \sum_{i = 1}^{n}\left \lceil \log_2{(i)} \right \rceil = n\left \lceil \log_2{(n)} \right \rceil - ...
1
vote
1answer
86 views

$\sum_{p\le x} \frac{1}{pq}$

I was given that $\sum_{p\le x} \frac{1}{p}$ = $\log\log x$+O(1). I need to show that $\sum_{pq\le x} \frac{1}{pq} = (\log \log x)^2 + O(\log \log x)$. Here we go: Break the sum into two sums: ...
6
votes
2answers
71 views

Is this summation solvable? $S_n = \sum_{i = 1}^{n}\log_i{(n)}$

Is it possible to solve a summation with a variable base of log? $$ S_n = \sum_{i = 2}^{n}\log_i{(n)} $$ Should I use the derivative of $\log_i{(n)}$?
0
votes
2answers
31 views

Approximating the sum of integers with the logarithm

Why does the following hold? $\sum_{j=1}^{n-1}j \to \log(n) \text{ as } n \to \infty$ Thanks!
0
votes
0answers
25 views

Summation of a function with the variable both in the function amd in the upper limit

E is defined as : E = c1 ( a$\rho$ + b$\rho ^{2}$ ) + c2 $\rho$ ( c + d $\sum_{j=0}^{n} (\log{ \frac{R\rho}{j} } ) $ ) + c3 $\rho ^{2}$ a, b, c, d, c1, c2, c3, R are known constants. $\rho$ is the ...
1
vote
1answer
38 views

What is the reasoning behind ways of splitting up this summation sign?

Some context: I've been studying Chebyshev's $\psi$ - function, which claims that $\psi(x) = \sum_{n \le x} \Lambda(n) = \sum_{p^k \le x} \log p$ where $p$ is prime and $\Lambda(n)$ is the von ...
0
votes
0answers
40 views

Approximation for the logarithm of a summatory

I would like to find an approximation for: $$ \log \left(\sum_{i=1}^{N} a_i\exp(-b_i^2)\exp(-c_i^2)\right) $$ with $$ a_i = \frac{1}{\sqrt{(e^2 + e_i^2)(g^2 + g_i^2)}} \\ b_i = \frac{b-d_i}{2(e^2 + ...
0
votes
1answer
43 views

Only root of a sum?

I have the following equation: $\sum\limits_{i=1}^{k}\left(n_{i}-n\cdot p_{i}\right)\log p_{i}=0$ where $p_{1},p_{2},...,p_{k}$ are the unknown variables with the condition: ...
0
votes
2answers
80 views

$\sum_{i=1}^n 1/i \leq c\log n$

This is what I want to show: $\sum_{i=0}^n 1/i \leq c \log n$ for all $n>N$ My current approach was this: $\sum_{i=1}^n 1/i = ( \int \sum_{i=1}^n 1/i )' = ( \sum_{i=1}^n \int 1/i )' = ( ...
1
vote
2answers
57 views

Logarithmic approximation of $\sum_0^{N-1} \frac{1}{2i + 1}$

Can anyone confirm that it's possible to approximate the sum $\sum_0^{N-1} \frac{1}{2i + 1}$ with the $\frac{\log{N}}{2}$? And why? It's clearly visible that the sum has a logarithmic growth over ...
0
votes
1answer
21 views

Make derivations over sums

I have this kind of sums $$ \left(\sum_{i_1=0}^{4}\sum_{i_2=0}^{4}\log(f(X,i_1,i_2))\right)'\ $$ And we want to derive in respect to $${x_i}$$, which is an element of the vector X. How I should do ...
1
vote
0answers
58 views

Estimation of a logarithmic sum

I need to estimate the sum $$ \underset{r=2}{\overset{t}{\sum}}\left(\frac{\log\log r}{r}\right)^{2}. $$ I tried to use the Abel's partial summation, and I got $$ \frac{(\log\log ...
3
votes
2answers
497 views

What's the formula to solve summation of logarithms?

this is my first question here. I'm studying summation and everything I know is that: $\sum_{i=1}^n\ k$ is $\frac{n(n+1)}{2}\ $ $\sum_{i=1}^{n}\ k^2$ is $\frac{n(n+1)(2n+1)}{6}\ $ $\sum_{i=1}^{n}\ ...
4
votes
3answers
105 views

Best approximation for $\displaystyle \sum_{k=2}^n\ln\ln k$?

I need the best approximation for $\displaystyle{\sum_{k = 2}^{n}\ln\left(\ln\left(k\right)\right)}$. Any suggestion or hint is welcomed. I derived $n\ln\left(\ln\left(n!\right) \over n\right)$ so ...
0
votes
1answer
62 views

How do I go about manipulating this summation equation to solve it?

In my textbook, Introduction to Algorithms, the following is shown: And I believe I understand that. However, I have a similar equation to the one on the first line, but instead of ...
0
votes
1answer
63 views

From $ \sum^\infty_{\lfloor \log n \rfloor + 1}n/{2^r} $ to $ \sum^\infty_{r=0}1/2^r $?

$$ E[h] = E[\sum^\infty_{r=1}I_r] = \sum^\infty_{r=1}E[I_r] $$ $$ = \sum^{ \lfloor \log n \rfloor}_{r=1}E[I_r] + \sum^\infty_{\lfloor \log n \rfloor + 1}E[I_r] $$ $$ \leq \sum^{ \lfloor \log n ...
0
votes
0answers
67 views

Representation of logarithm of $n$

What is the nearest representation of $\log(n)$ using $n$ when $n$ is an integer ? I found these definitions of logarithm : $$\log\left(1-x\right) = - \sum^{\infty}_{n=1} \frac{x^n}n\quad\text{ for ...
0
votes
2answers
104 views

How to go from a sum to a product and a product to a sum?

I have read here (third post down) that exponentials turn sums into products and logarithms turn products into sums. Can someone please further explain this?
0
votes
3answers
259 views

Ln Series Summation

I have been given: $$\ln{n}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\text{ for sufficiently large }n$$ Which I can equate to $\ln(n)=\sum\limits_{i=1}^n \frac{1}{i}$ The series I need ...
5
votes
4answers
665 views

Summation of logs

Are there any useful identities for quickly calculating the sum of consecutive logs? For example $\sum_{k=1}^{N} log(k)$ or something to this effect. I should add that I am writing code to do this (as ...
3
votes
1answer
79 views

What is the closed formula for the following summation?

Is there any closed formula for the following summation? $$\sum_{k=2}^n \frac{1}{\log_2(k)}$$
3
votes
2answers
159 views

Convergence of integral of log and sum the mean of the logs

How can I show that the following limit converges and $L \in (0, +\infty)$? $\lim\limits_{n \to +\infty}\left( S_n - T_n\right)$, where $S_n = \int\limits_1^n \log x\, dx$, and $T_n = \sum_{k = ...
5
votes
3answers
199 views

Why is $\int\limits_{1}^{n} \log x \,dx \le \sum\limits_{x = 1}^{n}\log x$?

It has been a long time since I studied integrals, so this question may sound stupid. I was going through this wiki page, and came across the following inequality: $$\int_{1}^{n} \log x \,dx \le ...
2
votes
1answer
390 views

Natural log summation representation

In working through a problem, I've encountered the need to express $\log n = \sum \limits_{k = 1}^{n - 1} \log(1 + \frac{1}{k})$ where $\log k $ is the natural logarithm of k. I'm fairly certain the ...
5
votes
2answers
2k views

Value of Summation of log(i)

Context: I am learning Dijstra's Algorithm to find shortest path to any node, given the start node. Here, we can use Fibonnacci Heap as Priority Queue. Following is few lines of algorithm: ...
0
votes
2answers
53 views

Proving an identity involving $f(n) = \sum_1^n{\lceil{\log_2 k}\rceil}$

Prove that $ f(n) = n - 1 + f(\lfloor {n/2} \rfloor) + f(\lceil n/2 \rceil)$ where $f(n) = \sum_1^n{\lceil{\log_2 k}\rceil}$ My trials: At first I find a formula for $f(n)$. If $n = 2^m$ , then ...
1
vote
1answer
83 views

Evaluate $\sum_{n=1}^{1024}\left \lfloor \log_2n \right \rfloor$.

Evaluate $\sum_{n=1}^{1024}\left \lfloor \log_2n \right \rfloor$. I thought the answer is $1+1*2+2*2^2+3*2^3+4*2^4+5*2^5+6*2^6+7*2^7+8*2^8+9*2^9+2^{10}=9219$, but the answer should be 8204. What ...