# តារាងនៃស៊េរីគណិតវិទ្យា

## ផលបូកនៃស្វ័យគុណ

• ${\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}}$
• ${\displaystyle \sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}={\frac {n^{3}}{3}}+{\frac {n^{2}}{2}}+{\frac {n}{6}}}$
• ${\displaystyle \sum _{i=1}^{n}i^{3}=\left({\frac {n(n+1)}{2}}\right)^{2}={\frac {n^{4}}{4}}+{\frac {n^{3}}{2}}+{\frac {n^{2}}{4}}=\left[\sum _{i=1}^{n}i\right]^{2}}$
• ${\displaystyle \sum _{i=1}^{n}i^{4}={\frac {n(n+1)(2n+1)(3n^{2}+3n-1)}{30}}}$
• ${\displaystyle \sum _{i=0}^{n}i^{s}={\frac {(n+1)^{s+1}}{s+1}}+\sum _{k=1}^{s}{\frac {B_{k}}{s-k+1}}{s \choose k}(n+1)^{s-k+1}}$
ដែល ${\displaystyle B_{k}}$ ជាចំនួនប៊ែរនូយី(Bernoulli number)ទីk
• ${\displaystyle \sum _{i=1}^{\infty }i^{-s}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}=\zeta (s)}$
ដែល ${\displaystyle \zeta (s)}$ជាអនុគមន៍ហ្សេតារីម៉ាន(Reimann zeta function) ។

## ស៊េរីស្វ័យគុណ

ផលបូកអនន្ត (ចំពោះ ${\displaystyle |x|\leq 1,x\neq 1}$)       ផលបូកមិនអនន្ត
${\displaystyle \sum _{i=0}^{\infty }x^{i}={\frac {1}{1-x}}}$ ${\displaystyle \sum _{i=0}^{n}x^{i}={\frac {1-x^{n+1}}{1-x}}}$
${\displaystyle \sum _{i=1}^{\infty }ix^{i}={\frac {x}{(1-x)^{2}}}}$ ${\displaystyle \sum _{i=1}^{n}ix^{i}=x{\frac {1-x^{n}}{(1-x)^{2}}}-{\frac {nx^{n+1}}{1-x}}}$
${\displaystyle \sum _{i=1}^{\infty }i^{2}x^{i}={\frac {x(1+x)}{(1-x)^{3}}}}$ ${\displaystyle \sum _{i=1}^{n}i^{2}x^{i}=}$ ${\displaystyle {\frac {x(1+x-(n+1)^{2}x^{n}+(2n^{2}+2n-1)x^{n+1}-n^{2}x^{n+2})}{(1-x)^{3}}}}$
${\displaystyle \sum _{i=1}^{\infty }i^{3}x^{i}={\frac {x(1+4x+x^{2})}{(1-x)^{4}}}}$
${\displaystyle \sum _{i=1}^{\infty }i^{4}x^{i}={\frac {x(1+x)(1+10x+x^{2})}{(1-x)^{5}}}}$
${\displaystyle \sum _{i=1}^{\infty }i^{k}x^{i}=x{\frac {d}{dx}}\left(\sum _{i=1}^{\infty }i^{\left(k-1\right)}x^{i}\right)=Li_{-k}(x)}$ ដែល ${\displaystyle Li_{s}(x)}$ ជាពហុលោការីត(Polylogarithm)នៃ ${\displaystyle x}$.

### ភាគបែងធម្មតា

• ${\displaystyle \sum _{i=1}^{\infty }{\frac {x^{i}}{i}}=\log _{e}\left({\frac {1}{1-x}}\right)\!}$ ចំពោះ${\displaystyle |x|\leq 1,\,x\not =-1}$
• ${\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}}{2i+1}}x^{2i+1}=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots =\arctan(x)}$
• ${\displaystyle \sum _{i=0}^{\infty }{\frac {x^{2i+1}}{2i+1}}=\mathrm {arctanh} (x)\!}$ចំពោះ${\displaystyle |x|<1\!}$

### ភាគបែងមានហ្វាក់តូរ្យែល(Factorial denominators)

ស៊េរីជាច្រើនដែលកើតឡើងពីទ្រឹស្តីបទតាយលើ(Taylor's theorem)​មានមេគុណដែលមានហ្វាក់តូរ្យែល ។

• ${\displaystyle \sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}}$
• ${\displaystyle \sum _{i=0}^{\infty }i{\frac {x^{i}}{i!}}=xe^{x}}$ (c.f. មធ្យមនៃរបាយព័យសុង(Poisson distribution))
• ${\displaystyle \sum _{i=0}^{\infty }i^{2}{\frac {x^{i}}{i!}}=(x+x^{2})e^{x}}$ (c.f. ម៉ូម៉ង់ទី២នៃរបាយព័យសុង)
• ${\displaystyle \sum _{i=0}^{\infty }i^{3}{\frac {x^{i}}{i!}}=(x+3x^{2}+x^{3})e^{x}}$
• ${\displaystyle \sum _{i=0}^{\infty }i^{4}{\frac {x^{i}}{i!}}=(x+7x^{2}+6x^{3}+x^{4})e^{x}}$

• ${\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i+1)!}}x^{2i+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots =\sin x}$
• ${\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i)!}}x^{2i}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots =\cos x}$
• ${\displaystyle \sum _{i=0}^{\infty }{\frac {x^{2i+1}}{(2i+1)!}}=\sinh x}$
• ${\displaystyle \sum _{i=0}^{\infty }{\frac {x^{2i}}{(2i)!}}=\cosh x}$

### ភាគបែងមានហ្វាក់តូរ្យែលសាំញ៉ាំ(Modified-factorial denominators)

• ${\displaystyle \sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}=\arcsin x\!}$ ចំពោះ${\displaystyle |x|<1\!}$
• ${\displaystyle \sum _{i=0}^{\infty }{\frac {(-1)^{i}(2i)!}{4^{i}(i!)^{2}(2i+1)}}x^{2i+1}=\mathrm {arsinh} (x)\!}$ ចំពោះ ${\displaystyle |x|<1\!}$

### ស៊េរីទ្វេធា(Binomial series)

ស៊េរីទ្វេធា (រួមទាំងរឹសការេនៃ ${\displaystyle \alpha =1/2}$ ហើយស៊េរីធរណីមាត្រអនន្តចំពោះ ${\displaystyle \alpha =-1}$):

• ${\displaystyle {\sqrt {1+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)n!^{2}4^{n}}}x^{n}\!}$ ចំពោះ${\displaystyle |x|<1\!}$
• ${\displaystyle (1+x)^{-1}=\sum _{n=0}^{\infty }(-1)^{n}x^{n}\!}$ ចំពោះ${\displaystyle |x|<1}$

ទំរង់ទូទៅ:

• ${\displaystyle (1+x)^{\alpha }=\sum _{n=0}^{\infty }{\alpha \choose n}x^{n}\!}$ ចំពោះ${\displaystyle |x|<1\!}$ ហើយចំពោះគ្រប់ចំនួនកុំផ្លិច${\displaystyle \alpha \!}$
ដោយសំរួលមេគុណទ្វេធា
${\displaystyle {\alpha \choose n}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}\!}$
• [១] ${\displaystyle \sum _{i=0}^{\infty }{i+n \choose i}x^{i}={\frac {1}{(1-x)^{n+1}}}}$
• [១] ${\displaystyle \sum _{i=0}^{\infty }{\frac {1}{i+1}}{2i \choose i}x^{i}={\frac {1}{2x}}({\sqrt {1-4x}})}$
• [១] ${\displaystyle \sum _{i=0}^{\infty }{2i \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}}$
• [១] ${\displaystyle \sum _{i=0}^{\infty }{2i+n \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}\left({\frac {1-{\sqrt {1-4x}}}{2x}}\right)^{n}}$

## មេគុណទ្វេធា(Binomial coefficients)

• ${\displaystyle \sum _{i=0}^{n}{n \choose i}=2^{n}}$
• ${\displaystyle \sum _{i=0}^{n}{n \choose i}a^{(n-i)}b^{i}=(a+b)^{n}}$
• ${\displaystyle \sum _{i=0}^{n}{i \choose k}={n+1 \choose k+1}}$
• ${\displaystyle \sum _{i=0}^{n}{k+i \choose i}={k+n+1 \choose n}}$
• ${\displaystyle \sum _{i=0}^{r}{r \choose i}{s \choose n-i}={r+s \choose n}}$

## អនុគមន៍ត្រីកោណមាត្រ

ផលបូក នៃ ស៊ីនុស និងកូស៊ីនុស មានក្នុងស៊េរីហ្វូរៀរ(Fourier series)​ ។

• ${\displaystyle \sum _{i=1}^{n}\sin \left({\frac {i\pi }{n}}\right)=0}$
• ${\displaystyle \sum _{i=1}^{n}\cos \left({\frac {i\pi }{n}}\right)=0}$

## មិនមានចំណាត់ថ្នាក់

• ${\displaystyle \sum _{n=b+1}^{\infty }{\frac {b}{n^{2}-b^{2}}}=\sum _{n=1}^{2b}{\frac {1}{2n}}}$