តារាងនៃស៊េរីគណិតវិទ្យា

ផលបូកនៃស្វ័យគុណ

• $\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}$
• $\sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}={\frac {n^{3}}{3}}+{\frac {n^{2}}{2}}+{\frac {n}{6}}$
• $\sum _{i=1}^{n}i^{3}=\left({\frac {n(n+1)}{2}}\right)^{2}={\frac {n^{4}}{4}}+{\frac {n^{3}}{2}}+{\frac {n^{2}}{4}}=\left[\sum _{i=1}^{n}i\right]^{2}$
• $\sum _{i=1}^{n}i^{4}={\frac {n(n+1)(2n+1)(3n^{2}+3n-1)}{30}}$
• $\sum _{i=0}^{n}i^{s}={\frac {(n+1)^{s+1}}{s+1}}+\sum _{k=1}^{s}{\frac {B_{k}}{s-k+1}}{s \choose k}(n+1)^{s-k+1}$

ដែល $B_{k}$ ជាចំនួនប៊ែរនូយី(Bernoulli number)ទីk ។

• $\sum _{i=1}^{\infty }i^{-s}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}=\zeta (s)$

ដែល $\zeta (s)$ជាអនុគមន៍ហ្សេតារីម៉ាន(Reimann zeta function) ។

ស៊េរីស្វ័យគុណ

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

ភាគបែងធម្មតា

• $\sum _{i=1}^{\infty }{\frac {x^{i}}{i}}=\log _{e}\left({\frac {1}{1-x}}\right)\!$ ចំពោះ$|x|\leq 1,\,x\not =-1$

• $\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{2i+1}}x^{2i+1}=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots =\arctan(x)$

• $\sum _{i=0}^{\infty }{\frac {x^{2i+1}}{2i+1}}=\mathrm {arctanh} (x)\!$ចំពោះ$|x|<1\!$

ភាគបែងមានហ្វាក់តូរ្យែល(Factorial denominators)

ស៊េរីជាច្រើនដែលកើតឡើងពីទ្រឹស្តីបទតាយលើ(Taylor's theorem)​មានមេគុណដែលមានហ្វាក់តូរ្យែល ។

• $\sum _{i=0}^{\infty }{\frac {x^{i}}{i!}}=e^{x}$

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

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

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

• $\sum _{i=0}^{\infty }{\frac {x^{2i+1}}{(2i+1)!}}=\sinh x$

• $\sum _{i=0}^{\infty }{\frac {x^{2i}}{(2i)!}}=\cosh x$

ភាគបែងមានហ្វាក់តូរ្យែលសាំញ៉ាំ(Modified-factorial denominators)

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

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

ស៊េរីទ្វេធា(Binomial series)

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

រឹសការេ:

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

ស៊េរីធរណីមាត្រ:

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

ទំរង់ទូទៅ:

• $(1+x)^{\alpha }=\sum _{n=0}^{\infty }{\alpha \choose n}x^{n}\!$ ចំពោះ$|x|<1\!$ ហើយចំពោះគ្រប់ចំនួនកុំផ្លិច$\alpha \!$

ដោយសំរួលមេគុណទ្វេធា

${\alpha \choose n}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}\!$

• $\sum _{i=0}^{\infty }{i+n \choose i}x^{i}={\frac {1}{(1-x)^{n+1}}}$
• $\sum _{i=0}^{\infty }{\frac {1}{i+1}}{2i \choose i}x^{i}={\frac {1}{2x}}({\sqrt {1-4x}})$
• $\sum _{i=0}^{\infty }{2i \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}$
• $\sum _{i=0}^{\infty }{2i+n \choose i}x^{i}={\frac {1}{\sqrt {1-4x}}}\left({\frac {1-{\sqrt {1-4x}}}{2x}}\right)^{n}$

មេគុណទ្វេធា(Binomial coefficients)

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

អនុគមន៍ត្រីកោណមាត្រ

ផលបូក នៃ ស៊ីនុស និងកូស៊ីនុស មានក្នុងស៊េរីហ្វូរៀរ(Fourier series)​ ។

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

មិនមានចំណាត់ថ្នាក់

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