# Tagged Questions

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### Calculate $\sum\limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$

Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$ I use software to complete the series is $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$ I have no idea about it. :|
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### Partial summation: integral version

In a book about analytic number theory, I found two lemmas about partial summation. The first one is the discrete version of partial summation (See http://en.wikipedia.org/wiki/Summation_by_parts) ...
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### How to prove that homometric sets lead to same result in this problem? (any justifications?)

First let me define Difference multiset for a set of integers $$P=\{p_1,p_2, \dots,p_K\} ,\quad p_i \in\{1,2,\dots,N\},\quad p_i\ne p_j$$ as below: $$D = \{p_i-p_j \mod N ,\quad i \ne j\}$$ I ...
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### Generalized PNT in limit as numbers get large

If $\pi_k(n)$ is the cardinality of numbers with k prime factors (repetitions included) less than or equal n, the generalized Prime Number Theorem (GPNT) is: \pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln ...
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### Calculating the limit of the derivative of a sum of digamma functions

The (real) digamma function is new to me and I notice that under some circumstances a sum of $\log\Gamma$ functions will have a derivative consisting of a sum of digamma functions that converge to a ...
Is anything known about these problems? If we make a string S of 0's and 1's with 1 in n'th position if the the nth prime $p_n$ is of the form $1+m 2^{9^{9^{9^{9}}}}$, else 0, does every finite string ...
The prime number theorem (PNT) says that an integer $n$ is prime with probability $\frac{1}{\ln n}$. Using only PNT, it's conceivable that each integer upto $10^{10^{10}}$ is non-prime. However using ...