query: How to prove that : $\sum \limits_{k=0}^{n-1} \lfloor x+\frac{k}{n}\rfloor = \lfloor nx\rfloor$?
positive_passages.docid: 156831
positive_passages.text: Prove $\Sigma_{k=0}^{n-1}\lfloor x+\frac{k}{n}\rfloor=\lfloor nx\rfloor$ , n is a Natural Number
negative_passages.docid: 155377
negative_passages.text: How can I calculate this limit: $\lim\limits_{x\to 0} x\left\lfloor\frac{1}{x}\right\rfloor$?
negative_passages.docid: 107789
negative_passages.text: Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$
negative_passages.docid: 2885
negative_passages.text: Prove that $\sum\limits_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$
negative_passages.docid: 107258
negative_passages.text: Proving If $k \le \left\lfloor \frac{n}{2} \right\rfloor$ then $\binom{n}{k-1} < \binom{n}{k}$
negative_passages.docid: 123955
negative_passages.text: How to compute $\sum\limits_{k=0}^n (-1)^k{2n-k\choose k}$?
negative_passages.docid: 10777
negative_passages.text: Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$
negative_passages.docid: 74286
negative_passages.text: Prove: $\lim_{n\to\infty} \frac{\sum^n_{k=1} \frac{1}{k}}{ \ln(n) } = 1$
negative_passages.docid: 49771
negative_passages.text: Prove that $\sum\limits_{k=0}^r\binom{n+k}k=\binom{n+r+1}r$ using combinatoric arguments.
negative_passages.docid: 79836
negative_passages.text: Proof for $\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$
negative_passages.docid: 92046
negative_passages.text: Proving $\sum\limits_{i=0}^n  i 2^{i-1} = (n+1) 2^n - 1$ by induction
negative_passages.docid: 82803
negative_passages.text: Prove that $\lfloor\lfloor x/2 \rfloor / 2 \rfloor = \lfloor x/4 \rfloor$
negative_passages.docid: 49928
negative_passages.text: Proving limit of a sum
negative_passages.docid: 44328
negative_passages.text: Prove that $\sum_{k=1} ^ n k^{-1} = \ln(n) + O(1)$
negative_passages.docid: 141446
negative_passages.text: Proving by induction that $\sum\limits_{k=2}^n \frac1{k^2}\le 1$
negative_passages.docid: 158515
negative_passages.text: Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$
negative_passages.docid: 93813
negative_passages.text: How to show that $\sum\limits_{k=0}^n (-1)^k\tfrac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$