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r+r^{2}+\cdots +r^{n}\right)(1-r)&=\left((1-r)+(r-r^{2})+(r^{2}-r^{3})+\dots +(r^{n}-r^{n+1})\right)\\&=1+(-r+r)+(-r^{2}+r^{2})+\dots +(-r^{n}+r...

(stability) }}\\&=c+r\sum _{k=0}^{\infty }cr^{k}&&{\text{ (linearity) }}\\&=c+r\,G(r,c),&&{\text{ hence }}\\G(r,c)&={\frac {c}{1-r}},{\text{ unless it...

sum R ⊕ R {\displaystyle \mathbb {R} \oplus \mathbb {R} } , where R {\displaystyle \mathbb {R} } is real coordinate space, is the Cartesian plane, R 2...

In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula c q ( n ) = ∑...

defined by, M R = ∑ i m i R i . {\displaystyle M\mathbf {R} =\sum _{i}m_{i}\mathbf {R} _{i}.} By inspection, R i = R + r i {\displaystyle \mathbf {R} _{i}=\mathbf...

Dim sum (traditional Chinese: 點心; simplified Chinese: 点心; pinyin: diǎn xīn; Jyutping: dim2 sam1) is a large range of small Chinese dishes that are traditionally...

In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician...

{r}}}\sum _{j=0}^{r-1}|\psi _{j}\rangle &={\frac {1}{r}}\sum _{j=0}^{r-1}\sum _{k=0}^{r-1}\omega _{r}^{jk}|a^{k}\rangle \\&=|1\rangle +{\frac {1}{r}}\sum...

multiplication}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1...

Sumer (/ˈsuːmər/) is the earliest known civilization, located in the historical region of southern Mesopotamia (now south-central Iraq), emerging during...

) = ∏ i = 1 r ∑ j = 0 a i p i j x = ∏ i = 1 r ( 1 + p i x + p i 2 x + ⋯ + p i a i x ) . {\displaystyle \sigma _{x}(n)=\prod _{i=1}^{r}\sum _{j=0}^{a_{i}}p_{i}^{jx}=\prod...

r = ( 1 − q ) − r = ∑ k = 0 ∞ ( − r − k ) ( − q ) k = ∑ k = 0 ∞ ( k + r − 1 k ) q k {\displaystyle p^{-r}=(1-q)^{-r}=\sum _{k=0}^{\infty }{\binom {-r}{{\phantom...

Sum 41 is a Canadian rock band from Ajax, Ontario. The band was formed in 1996 and currently consists of Deryck Whibley (lead vocals, guitars, keyboards)...

}(k+r)A_{k}z^{k+r}+q(z)\sum _{k=0}^{\infty }A_{k}z^{k+r}\\={}&\sum _{k=0}^{\infty }[(k+r-1)(k+r)A_{k}z^{k+r}+p(z)(k+r)A_{k}z^{k+r}+q(z)A_{k}z^{k+r}]\\={}&\sum...

_{ir}(1)}{R}}} a i j ∗ = ∑ r = 1 R ∑ t = 1 T − 1 ξ i j r ( t ) ∑ r = 1 R ∑ t = 1 T − 1 γ i r ( t ) , {\displaystyle a_{ij}^{*}={\frac {\sum _{r=1}^{R}\sum _{t=1}^{T-1}\xi...

+ n r ) = ∑ k = 0 r ( m k ) ( n r − k ) {\displaystyle {m+n \choose r}=\sum _{k=0}^{r}{m \choose k}{n \choose r-k}} for any nonnegative integers r, m,...

n = r {\displaystyle n=r} ; ∑ i = r n ( i r ) = ∑ i = r r ( i r ) = ( r r ) = 1 = ( r + 1 r + 1 ) = ( n + 1 r + 1 ) . {\displaystyle \sum _{i=r}^{n}{i...

r ( a ) ( n − 1 ) ! ( v ) n , ℜ ( v ) > − a , r = 1 , 2 , 3 , … {\displaystyle \psi (v)={\frac {1}{r}}\sum _{l=0}^{r-1}\ln(v+a+l)+{\frac {1}{r}}\sum _{n=1}^{\infty...

sum _{i}x_{i}&&=\sum _{i}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{i<j}x_{i}x_{j}&&=\sum _{i<j}\tan \theta _{i}\tan \theta _{j}\\[6pt]e_{3}&=\sum...

theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically G ( χ ) := G ( χ , ψ ) = ∑ χ ( r ) ⋅ ψ ( r ) {\displaystyle...