query: \sum_{k=0, k even}^n {n \choose k}*2^k = \frac{(3^n)+((-1)^n)}{2}
positive_passages.docid: 156564
positive_passages.text: Prove the identity $\sum_{{{\underset{k-even}{k=0}}}}^{n}{n \choose k}2^{k}=\frac{3^{n}+(-1)^{n}}{2}$
negative_passages.docid: 114911
negative_passages.text: Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$
negative_passages.docid: 74117
negative_passages.text: Find the sum of series $\sum_{n=0}^{\infty}\frac{z^{kn}}{(kn)!}$
negative_passages.docid: 137249
negative_passages.text: find $\sum_{k=2}^{\infty}\frac{1}{k^2-1}$
negative_passages.docid: 47706
negative_passages.text: Find the value of $ \sum_{n=1}^{\infty} \frac{n^3}{3^n} $
negative_passages.docid: 326
negative_passages.text: Find $\sum_{k=1}^{\infty}\frac{1}{2^{k+1}-1}$
negative_passages.docid: 89698
negative_passages.text: Please help with $\sum\limits_{r=1}^{N}\frac{2r+3}{3^r(r+1)}$
negative_passages.docid: 118571
negative_passages.text: Why is $\sum\limits_{k=0}^{n}(-1)^k\binom{n}{k}^2=(-1)^{n/2}\binom{n}{n/2}$ if $n$ is even?
negative_passages.docid: 76828
negative_passages.text: Evaluating $\sum_{i = 0}^{\lfloor n/2 \rfloor} {n \choose 2i}p^{2i}(1 - p)^{n - 2i}$
negative_passages.docid: 58428
negative_passages.text: Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
negative_passages.docid: 37650
negative_passages.text: Find $\sum_{m=0}^n\ (-1)^m m^n {n \choose m}$
negative_passages.docid: 134610
negative_passages.text: Find the sum of finite series $S=\sum_{k=1}^{2015}{(-1)^{\frac{k(k+1)}{2}}}k$
negative_passages.docid: 122534
negative_passages.text: Show that $\sum (-1)^{n+1}/n$ = $k$ can be expanded to a sum with $ 1 - 1/2 + 1/3 + 1/5 - 1/4 +1/7 + 1/9 - 1/6 + ....$ which is $3k/2$
negative_passages.docid: 4991
negative_passages.text: Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$
negative_passages.docid: 5272
negative_passages.text: Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$
negative_passages.docid: 5054
negative_passages.text: Calculate $\sum\limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$
negative_passages.docid: 17638
negative_passages.text: Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$