query: How to proof the following identity: $ \sum_{k=0}^{n} \prod_{j=0\\j\neq k}^{n} \frac{t_k}{t_k-t_j} =1$
positive_passages.docid: 25882
positive_passages.text: Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$
negative_passages.docid: 50972
negative_passages.text: Prove that $\sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$
negative_passages.docid: 82742
negative_passages.text: Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $
negative_passages.docid: 16143
negative_passages.text: How can I prove that $\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1$?
negative_passages.docid: 32791
negative_passages.text: Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$
negative_passages.docid: 18046
negative_passages.text: Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
negative_passages.docid: 142666
negative_passages.text: Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$
negative_passages.docid: 114418
negative_passages.text: A deduction from $x^n-1=(x^2-1)\prod_{k=1}^{(n-2)/2}[x^2-2x\cos{\frac{2k\pi}{n}}+1]$
negative_passages.docid: 99249
negative_passages.text: Sum of product of binomial coefficients: $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{n + k}{k} = (-1)^n$
negative_passages.docid: 15164
negative_passages.text: Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
negative_passages.docid: 91081
negative_passages.text: Prove that, for every natural n, $\sum_{d|n} \frac{\mu(d)}{d}=\prod_{p|n}(1-\frac{1}{p})$
negative_passages.docid: 152104
negative_passages.text: How do I prove that $ij = k?$
negative_passages.docid: 44621
negative_passages.text: Simple binomial theorem proof: $\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k$
negative_passages.docid: 128913
negative_passages.text: A proof of $\sum_{i=1}^{n}(-1)^{i-1}\binom{n}{i}\frac{1}{i}=\frac{1}{1}+\frac{1}{2}+....+\frac{1}{n}$
negative_passages.docid: 70128
negative_passages.text: Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
negative_passages.docid: 134152
negative_passages.text: Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$
negative_passages.docid: 118471
negative_passages.text: Prove that $1-\sum_{i=1}^n a_i < \prod_{i=1}^n(1-a_i) $