query: Proving $1\cdot1! + 2\cdot2! + 3\cdot3! + ... + k\cdot k! = (k+1)! - 1$
positive_passages.docid: 146305
positive_passages.text: Simplify sum of factorials with mathematical induction
negative_passages.docid: 155769
negative_passages.text: Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
negative_passages.docid: 74887
negative_passages.text: Combinatorial proof of the identity ${n+1\choose k+1}={k\choose k}+{k+1\choose k}+...+{n-1\choose k}+{n\choose k}$
negative_passages.docid: 87411
negative_passages.text: Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$
negative_passages.docid: 106981
negative_passages.text: Prove that $(a+1)(a+2)...(a+b)$ is divisible by $b!$
negative_passages.docid: 63553
negative_passages.text: Show that $\frac{1}{(n+1)!}(1+\frac{1}{(n+1)}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots)=\frac{1}{n!n}$
negative_passages.docid: 44438
negative_passages.text: Prove : $\frac{\cos(x_1) +\cos(x_2) +\cdots+\cos(x_{10})}{\sin(x_1) +\sin(x_2) +\cdots+\sin(x_{10})} \ge 3$
negative_passages.docid: 106888
negative_passages.text: Factorial identity $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+\cdots$
negative_passages.docid: 75910
negative_passages.text: Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$
negative_passages.docid: 17491
negative_passages.text: Proof by Induction: Solving $1+3+5+\cdots+(2n-1)$
negative_passages.docid: 82591
negative_passages.text: Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.
negative_passages.docid: 120954
negative_passages.text: Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
negative_passages.docid: 64378
negative_passages.text: Proving a binomial sum identity $\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}$
negative_passages.docid: 58589
negative_passages.text: On the factorial equations $A! B! =C!$ and $A!B!C! = D!$
negative_passages.docid: 68848
negative_passages.text: How to simplify $\sum_{k=1}^nk\cdot k!$
negative_passages.docid: 92704
negative_passages.text: Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
negative_passages.docid: 25961
negative_passages.text: Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$