Tagged Questions
5
votes
4answers
113 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
68 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
83 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
102 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 ...
1
vote
1answer
139 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 ...
3
votes
2answers
166 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
48 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 ...
