query: Closed form for $1 + 3 + 5 + \cdots +(2n-1)$
positive_passages.docid: 42831
positive_passages.text: How to derive the formula for the sum of the first $n$ odd numbers: $n^2=\sum_{k=1}^n(2k-1).$
negative_passages.docid: 92754
negative_passages.text: Closed form for the series: $\sum_{n=1}^{\infty}x^{n^2}$
negative_passages.docid: 28109
negative_passages.text: Conjectural closed-form representations of sums, products or integrals
negative_passages.docid: 121441
negative_passages.text: $1\cdot3\cdot5 \cdots (p - 2) = (-1)^{m+k+1} \pmod p$, and $2\cdot4\cdot6\cdots(p - 1) = ( -1)^{m +k} \pmod p$.
negative_passages.docid: 125981
negative_passages.text: Closed-Form Solution to Infinite Sum
negative_passages.docid: 129801
negative_passages.text: Finding a closed form for $\sum^{n}_{k=1} \frac{k}{(k+1)!} $
negative_passages.docid: 6539
negative_passages.text: Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$?
negative_passages.docid: 73191
negative_passages.text: Proving that $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots + \frac{1}{n(n+1)} = \frac{n}{n+1}$
negative_passages.docid: 151085
negative_passages.text: Simplify the sum $ \sum_{i=0}^{k}(-1)^i i \binom{n}{i} \binom{n}{k-i}$
negative_passages.docid: 30851
negative_passages.text: Find the summation $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \cdots$
negative_passages.docid: 7302
negative_passages.text: $\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?
negative_passages.docid: 37059
negative_passages.text: limit of the sum $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n} $
negative_passages.docid: 634
negative_passages.text: Closed form formula for $\sum\limits_{k=1}^n k^k$
negative_passages.docid: 2675
negative_passages.text: Proof $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
negative_passages.docid: 153409
negative_passages.text: Sum of $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{12}+\cdots$
negative_passages.docid: 74294
negative_passages.text: How to simplify $\sum_{i=1}^{k}\binom{n + i - 1}{i}$?
negative_passages.docid: 113891
negative_passages.text: The limit of a sum of the form $a_0 \sqrt n + a_1 \sqrt {n + 1} +\cdots + a_k \sqrt {n + k}$