User:Cornince

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Hiragana/Katakana: むすかまみずかわでにほん

English: tie construct by oneself Japan

妙子さん、スターヲルスを水曜日の夜中に見たいですか。

$\sum_{i=m}^{i_p} \! {}^p \ f(i) = \sum_{i_{p-1}=m}^{i_p} \, \sum_{i_{p-2}=m}^{i_{p-1}} \, \sum_{i_{p-3}=m}^{i_{p-2}} ... \sum_{i_2=m}^{i_3} \, \sum_{i_1=m}^{i_2} \, \sum_{i_0=m}^{i_1} f(i_0) \,$

$\sum_{i=m}^{i_0} \! {}^0 \ \mathit{f}(i) = \mathit{f}(i_0)\,$

$\sum_{i=m}^{i_1} \! {}^1 \ \mathit{f}(i) = \sum_{i_0=m}^{i_1} \mathit{f}(i_0)\,$

$\mathrm{If \ } \mathit{g}(n) = \sum_{i=m}^n \! {}^{-p} \, \mathit{f}(i), \ \mathrm{then \ } \mathit{f}(n) = \sum_{i=m}^n \! {}^{p} \, \mathit{g}(i). \,$

$\sum_{i=m}^{i_p} \! {}^p \ \mathit{f}(i) = \sum_{i_{p-1}=m}^{i_p} \left [ \sum_{i=m}^{i_{p-1}} \! {}^{p-1} \, \mathit{f}(i) \right ] \,$

$\sum_{i_{p-1}=m}^{i_p} \left [ \sum_{i=m}^{i_{p-1}} \! {}^{p-1} \, \mathit{f}(i) \right ] = \sum_{i=m}^{m} \! {}^{p-1} \, \mathit{f}(i) + \sum_{i=m}^{m+1} \! {}^{p-1} \, \mathit{f}(i) + \sum_{i=m}^{m+2} \! {}^{p-1} \, \mathit{f}(i) + \, ... + \, \sum_{i=m}^{i_p - 2} \! {}^{p-1} \, \mathit{f}(i) + \sum_{i=m}^{i_p - 1} \! {}^{p-1} \, \mathit{f}(i) + \sum_{i=m}^{i_p} \! {}^{p-1} \, \mathit{f}(i) \,$

$\sum_{i=m}^{n} \mathit{f}(i) = \sum_{i=m+u}^{n+u} \mathit{f}(i - u) \,$

$\sum_{i=m}^{n} \mathit{f}(i) = \mathit{f}(m) + \mathit{f}(m+1) + \mathit{f}(m+2) + \, ... \, + \mathit{f}(n-2) + \mathit{f}(n-1) + \mathit{f}(n) \,$

$\,\! \begin{array}{lcl} \sum_{i=m+u}^{n+u} \mathit{f}(i - u) & = & \mathit{f}((m+u) - u) + \mathit{f}((m+u+1) - u) + \mathit{f}((m+u+2) - u) \\ & & + \, ... \, + \mathit{f}((n+u-2) - u) + \mathit{f}((n+u-1) - u) + \mathit{f}((n+u) - u) \\ \\ & = & \mathit{f}(m) + \mathit{f}(m+1) + \mathit{f}(m+2) + \, ... \, + \mathit{f}(n-2) + \mathit{f}(n-1) + \mathit{f}(n) \\ \\ & = & \sum_{i=m}^{n} \mathit{f}(i) \end{array} \,$

$\,\! \sum_{i=m}^{n} \! {}^p \ \mathit{f}(i) = \textstyle \! {}_p \! \sum_{i=m}^n \mathit{f}(i)\,$

$\,\! \begin{array}{lcl} {}_p \! \sum_{i=m}^n r^i & = & {r^{n+p} \over (r-1)^p} - \sum_{k=0}^{p-1} {r^{m + p - (k+1)} \prod_{j=1}^k (n - m + j) \over k! (r-1)^{p-k}} \\ & = & {r^{n+p} \over (r-1)^p} - \sum_{k=0}^{p-1} {r^{m + p - (k+1)} {(n - m + k)! \over (n - m)!} \over (r-1)^{p-k} k!} \\ & = & {r^{n+p} \over (r-1)^p} - \sum_{k=0}^{p-1} \left ({r^{m + p - (k+1)} \over (r-1)^{p-k}} \right ) {{n - m + k} \choose k} \end{array}\,$

$\mathrm{For\ natural\ numbers\ } q\ \mathrm{and\ } t \ge q, \quad \sum_{z=q}^{t} {z \choose q} = {t + 1 \choose q + 1}. \,$

$t = q, \quad \sum_{z=q}^{q} {z \choose q} = {q \choose q} = 1 = {q + 1 \choose q + 1} \,$

$t = c, \quad \sum_{z=q}^{c} {z \choose q} = {c + 1 \choose q + 1} \,$

$t = c + 1, \quad \sum_{z=q}^{c+1} {z \choose q} = \sum_{z=q}^{c} {z \choose q} + {c + 1 \choose q} = {c + 1 \choose q + 1} + {c + 1 \choose q} = {(c + 1) + 1 \choose q + 1} \,$

$\mathrm{For \ a\ real\ number\ } r \ne 1\ \mathrm{and\ natural\ numbers\ } m,\ p,\ \mathrm{and\ } i_p,\ \mathrm{where\ } i_p \ge m,$

$\sum_{i=m}^{i_p} \!\! {}^p \, r^i = {r^{i_p+p} \over (r-1)^p} - \sum_{k=0}^{p-1} {r^{m + p - (k+1)} \prod_{j=1}^k (i_p - m + j) \over k! (r-1)^{p-k}}. \,$

$p = 0, \quad \sum_{i=m}^{i_0} \!\! {}^0 \, r^i = r^{i_0} \,$

$p = 1, \quad \sum_{i=m}^{i_1} \!\! {}^1 \, r^i = {r^{i_1+1} \over (r-1)^1} - {r^m \over 0! (r-1)^1} = {r^{i_1+1} - r^{m} \over r-1} \,$

$p = 2, \quad \sum_{i=m}^{i_2} \!\! {}^2 \, r^i = {r^{i_2+2} \over (r-1)^2} - {r^{m+1} \over 0! (r-1)^2} - {r^m (i_2 - m + 1) \over 1! (r-1)^1} = {{r^{i_2+2} - r^{m+1} \over r-1} - r^m (i_2 - m + 1) \over r-1} \,$

$p = 3, \,$

$\quad \sum_{i=m}^{i_3} \!\! {}^3 \, r^i = {r^{i_3+3} \over (r-1)^3} - {r^{m+2} \over 0! (r-1)^3} - {r^{m+1} (i_3 - m + 1) \over 1! (r-1)^2} - {r^m (i_3 - m + 1)(i_3 - m + 2) \over 2! (r-1)^1} \,$

$= {{{r^{i_3+3} - r^{m+2} \over r-1} - r^{m+1} (i_3 - m + 1) \over r-1} - \left ( {1 \over 2} \right ) r^m (i_3 - m + 1)(i_3 - m + 2) \over r-1} \,$

$p = 4, \,$

$\quad \sum_{i=m}^{i_4} \!\! {}^4 \, r^i = {r^{i_4+4} \over (r-1)^4} - {r^{m+3} \over 0! (r-1)^4} - {r^{m+2} (i_4 - m + 1) \over 1! (r-1)^3} - {r^{m+1} (i_4 - m + 1)(i_4 - m + 2) \over 2! (r-1)^2} \,$

$- {r^{m} (i_4 - m + 1)(i_4 - m + 2)(i_4 - m + 3) \over 3! (r-1)^1} \,$

$= {{{{r^{i_4+4} - r^{m+3} \over r-1} - r^{m+2} (i_4 - m + 1) \over r-1} - \left ( {1 \over 2} \right ) r^{m+1} (i_4 - m + 1)(i_4 - m + 2) \over r-1} - \left ( {1 \over 6} \right ) r^{m} (i_4 - m + 1)(i_4 - m + 2)(i_4 - m + 3) \over r-1} \,$

$p = a, \quad \sum_{i=m}^{i_a} \!\! {}^a \, r^i = {r^{{i_a}+a} \over (r-1)^a} - \sum_{k=0}^{a-1} {r^{m + a - (k+1)} \prod_{j=1}^k (i_a - m + j) \over k! (r-1)^{a-k}} \,$

$p = a+1, \quad \sum_{i=m}^{i_{(a+1)}} \!\! {}^{a+1} \, r^i = \sum_{{i_a}=m}^{i_{(a+1)}} \left [ \sum_{i=m}^{i_a} \!\! {}^a \, r^i \right ] \,$

$= \sum_{{i_a}=m}^{i_{(a+1)}} \left [ {r^{{i_a}+a} \over (r-1)^a} - \sum_{k=0}^{a-1} {r^{m + a - (k+1)} \prod_{j=1}^k (i_a - m + j) \over k! (r-1)^{a-k}} \right ] \,$

$= \sum_{{i_a}=m}^{i_{(a+1)}} \left [ {r^{{i_a}+a} \over (r-1)^a} \right ] - \sum_{{i_a}=m}^{i_{(a+1)}} \left [ \sum_{k=0}^{a-1} {r^{m + a - (k+1)} \prod_{j=1}^k (i_a - m + j) \over k! (r-1)^{a-k}} \right ] \,$

$= {\sum_{{i_a}=m}^{i_{(a+1)}} \left ( r^{{i_a}+a} \right ) \over (r-1)^a} - \sum_{{i_a}=m}^{i_{(a+1)}} \left [ \sum_{k=0}^{a-1} {r^{m + a - (k+1)} \left [{(i_a - m + k)! \over (i_a - m)!} \right ] \over (r-1)^{a-k} \qquad k!} \right ] \,$

$= {r^a \sum_{{i_a}=m}^{i_{(a+1)}} \left ( r^{i_a} \right ) \over (r-1)^a} - \sum_{{i_a}=m}^{i_{(a+1)}} \left [ \sum_{k=0}^{a-1} \left ( {r^{m + a - (k+1)} \over (r-1)^{a-k}} \right ) {i_a - m + k \choose k} \right ] \,$

$= {r^a \left ( {r^{i_{(a+1)} + 1} - r^m \over r-1} \right ) \over (r-1)^a} - \sum_{k=0}^{a-1} \left [ \left ( {r^{m + a - (k+1)} \over (r-1)^{a-k}} \right ) \sum_{{i_a}=m}^{i_{(a+1)}} {i_a - m + k \choose k} \right ] \,$

$= {r^{i_{(a+1)} + a + 1} - r^{m+a} \over (r-1)^{a+1}} - \sum_{k=0}^{a-1} \left [ \left ( {r^{m + a - (k+1)} \over (r-1)^{a-k}} \right ) \sum_{i_a=k}^{i_{(a+1)} - m + k} {i_a \choose k} \right ] \,$

$= {r^{i_{(a+1)} + a + 1} \over (r-1)^{a+1}} - {r^{m+a} \over (r-1)^{a+1}} - \sum_{k=0}^{a-1} \left ( {r^{m + a - (k+1)} \over (r-1)^{a-k}} \right ) {i_{(a+1)} - m + k + 1 \choose k + 1} \,$

$= {r^{i_{(a+1)} + a + 1} \over (r-1)^{a+1}} - {r^{m+a} \over (r-1)^{a+1}} - \sum_{k=1}^{(a+1)-1} \left ( {r^{m + a - ((k - 1)+1)} \over (r-1)^{a-(k - 1)}} \right ) {i_{(a+1)} - m + (k - 1) + 1 \choose (k - 1) + 1} \,$

$= {r^{i_{(a+1)} + a + 1} \over (r-1)^{a+1}} - {r^{m+a} \over (r-1)^{a+1}} - \sum_{k=1}^{(a+1)-1} \left ( {r^{m + a - k + 1 - 1} \over (r-1)^{a - k + 1}} \right ) {i_{(a+1)} - m + k \choose k} \,$

$= {r^{i_{(a+1)} + (a + 1)} \over (r-1)^{(a+1)}} - {r^{m+a} \over (r-1)^{a+1}} - \left [ \sum_{k=0}^{(a+1)-1} \left ( {r^{m + (a + 1) - k - 1} \over (r-1)^{(a + 1) - k}} \right ) {i_{(a+1)} - m + k \choose k} - \left ( {r^{m+a} \over (r-1)^{a+1}} \right ) {i_{(a+1)} - m \choose 0} \right ] \,$

$= {r^{i_{(a+1)} + (a + 1)} \over (r-1)^{(a+1)}} - \sum_{k=0}^{(a+1)-1} \left ( {r^{m + (a + 1) - (k + 1)} \over (r-1)^{(a + 1) - k}} \right ) {i_{(a+1)} - m + k \choose k} \,$

$= {r^{i_{(a+1)} + (a + 1)} \over (r-1)^{(a+1)}} - \sum_{k=0}^{(a+1)-1} {r^{m + (a + 1) - (k+1)} \left [{(i_{(a+1)} - m + k)! \over (i_{(a+1)} - m)!} \right ] \over (r-1)^{(a+1)-k} \qquad k!} \,$

$\sum_{i=m}^{i_{(a+1)}} \!\! {}^{a+1} \, r^i = {r^{i_{(a+1)} + (a + 1)} \over (r-1)^{(a+1)}} - \sum_{k=0}^{(a+1)-1} {r^{m + (a+1) - (k+1)} \prod_{j=1}^k (i_{(a+1)} - m + j) \over k! (r-1)^{(a+1)-k}} \,$

$\mathrm{Where\ integer\ } p \ge 1, \quad \sum_{i=m}^{i_p} \! {}^p \ f(i) = \sum_{i=m}^{i_p} {i_p - i + p - 1 \choose p - 1} f(i) \,$

$p = 1, \quad \sum_{i=m}^{i_1} \! {}^1 \ f(i) = \sum_{i=m}^{i_1} {i_p - i \choose 0} f(i) = \sum_{i=m}^{i_1} f(i) \,$

$p = s, \quad \sum_{i=m}^{i_s} \! {}^s \ f(i) = \sum_{i=m}^{i_s} {i_s - i + s - 1 \choose s - 1} f(i) \,$

$p = s + 1, \quad \sum_{i=m}^{i_{s+1}} \! {}^{s+1} \ f(i) = \sum_{i_s=m}^{i_{s+1}} \left [ \sum_{i=m}^{i_s} \! {}^s \, \mathit{f}(i) \right ] \,$

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User:Cornince

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