តារាងនៃស៊េរីគណិតវិទ្យា

ផលបូកនៃស្វ័យគុណ

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

ដែល $B_k$ ជាចំនួនប៊ែរនូយី(Bernoulli number)ទីk ។

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

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

ស៊េរីស្វ័យគុណ

ផលបូកអនន្ត (ចំពោះ $|x|\le 1,x\ne 1$)		ផលបូកមិនអនន្ត
$\sum _{i=0}^{\mathrm{\infty }}{x}^i=\frac{1}{1-x}$		$\sum _{i=0}^n{x}^i=\frac{1-x^{n+1}}{1-x}$
$\sum _{i=1}^{\mathrm{\infty }}i{x}^i=\frac{x}{(1-x{)}^2}$		$\sum _{i=1}^{n}i{x}^i=x\frac{1-x^n}{(1-x{)}^2}-\frac{n{x}^{n+1}}{1-x}$
$\sum _{i=1}^{\mathrm{\infty }}{i}^2{x}^i=\frac{x(1+x)}{(1-x{)}^3}$		$\sum _{i=1}^n{i}^2{x}^i=$ $\frac{x(1+x-(n+1{)}^2{x}^n+(2n^2+2n-1)x^{n+1}-n^2{x}^{n+2})}{(1-x{)}^3}$
$\sum _{i=1}^{\mathrm{\infty }}{i}^3{x}^i=\frac{x(1+4x+x^2)}{(1-x{)}^4}$
$\sum _{i=1}^{\mathrm{\infty }}{i}^4{x}^i=\frac{x(1+x)(1+10x+x^2)}{(1-x{)}^5}$
$\sum _{i=1}^{\mathrm{\infty }}{i}^k{x}^i=x\frac{d}{dx}\left(\sum _{i=1}^{\mathrm{\infty }}{i}^{(k-1)}{x}^i\right)=L{i}_{-k}(x)$ ដែល $L{i}_s(x)$ ជាពហុលោការីត(Polylogarithm)នៃ $x$.

ភាគបែងធម្មតា

• $\sum _{i=1}^{\mathrm{\infty }}\frac{x^i}{i}={\log }_e\left(\frac{1}{1-x}\right)$ ចំពោះ$|x|\le 1,x=-1$

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

• $\sum _{i=0}^{\mathrm{\infty }}\frac{x^{2i+1}}{2i+1}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(x)$ចំពោះ$|x|<1$

ភាគបែងមានហ្វាក់តូរ្យែល(Factorial denominators)

ស៊េរីជាច្រើនដែលកើតឡើងពីទ្រឹស្តីបទតាយលើ(Taylor's theorem)​មានមេគុណដែលមានហ្វាក់តូរ្យែល ។

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

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

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

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

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

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

ភាគបែងមានហ្វាក់តូរ្យែលសាំញ៉ាំ(Modified-factorial denominators)

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

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

ស៊េរីទ្វេធា(Binomial series)

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

រឹសការេ:

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

ស៊េរីធរណីមាត្រ:

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

ទំរង់ទូទៅ:

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

ដោយសំរួលមេគុណទ្វេធា

$(\genfrac{}{}{0ex}{}{\alpha }{n})=\prod _{k=1}^n\frac{\alpha -k+1}{k}=\frac{\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}$

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

មេគុណទ្វេធា(Binomial coefficients)

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

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

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