# Tagged Questions

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### 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 $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109$$ ...
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### Does $\sum_{i=1}^{k-1}\lceil \log_2\frac{N}{i}\rceil$ have a closed form?

Does the following have a closed formula? $$\sum_{i=1}^{k-1}\left\lceil \log_2\frac{N}{i}\right\rceil$$
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### 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 ...
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### 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 ...
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### 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 ...
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### Value of Summation of $\log(n)$

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: ...
### 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 ...
### 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 ...