query: If $n \in \mathbb N$, find $\sum(-1)^r \binom{n}{r}\left(\frac {1} {2^r}+\frac {3^r} {2^{2r}}+\frac {7^r} {2^{3r}} + \cdots \text{m terms}\right)$
positive_passages.docid: 150355
positive_passages.text: Finding sum of binomial series
negative_passages.docid: 116722
negative_passages.text: Simplifying $\big(\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}\big)/{\binom{M}{n}}$
negative_passages.docid: 57418
negative_passages.text: How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$
negative_passages.docid: 14399
negative_passages.text: Calculate $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k}$
negative_passages.docid: 92704
negative_passages.text: Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
negative_passages.docid: 39199
negative_passages.text: closed form for $\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}$
negative_passages.docid: 51864
negative_passages.text: $\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \right) = -1$
negative_passages.docid: 130543
negative_passages.text: If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
negative_passages.docid: 37650
negative_passages.text: Find $\sum_{m=0}^n\ (-1)^m m^n {n \choose m}$
negative_passages.docid: 86426
negative_passages.text: calculate $\sum_{1\le i\le r}\frac{i+1} { r+1}{2r-i\choose r-i}{s+i-2\choose i}+\frac{1}{r+1}{2r\choose r}$?
negative_passages.docid: 76319
negative_passages.text: How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?
negative_passages.docid: 145048
negative_passages.text: Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$
negative_passages.docid: 72639
negative_passages.text: Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
negative_passages.docid: 37569
negative_passages.text: Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $
negative_passages.docid: 112036
negative_passages.text: sum of series $\frac{n}{1\cdot 2 \cdot 3}+\frac{n-1}{2\cdot 3\cdot 4}+\frac{n-2}{3\cdot 4 \cdot 5}+\cdots \cdots n$ terms
negative_passages.docid: 562
negative_passages.text: Formula for calculating $\sum_{n=0}^{m}nr^n$
negative_passages.docid: 4327
negative_passages.text: proving $\left(1+\frac 1n\right)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem