មើលកូដរបស់ តារាងនៃស៊េរីគណិតវិទ្យា

==ផលបូកនៃស្វ័យគុណ==
* <math>\sum_{i=1}^n i = \frac{n(n+1)}{2}</math>
* <math>\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}</math>
* <math>\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</math>
* <math>\sum_{i=1}^{n} i^{4} = \frac{n(n+1)(2n+1)(3n^{2}+3n-1)}{30}</math>
* <math>\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}</math>
:ដែល <math>B_k</math> ជា[[ចំនួនប៊ែរនូយី]](Bernoulli number)ទី''k'' ។
* <math>\sum_{i=1}^\infty i^{-s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = \zeta(s)</math>
:ដែល <math>\zeta(s)</math>ជា[[អនុគមន៍ហ្សេតារីម៉ាន]](Reimann zeta function) ។

== ស៊េរីស្វ័យគុណ ==
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! ផលបូកអនន្ត (ចំពោះ <math>|x|\le 1, x\neq 1</math>) !! &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; !! ផលបូកមិនអនន្ត
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|<math>\sum_{i=0}^\infty x^i = \frac{1}{1-x}</math>
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|<math>\sum_{i=0}^n x^i = \frac{1-x^{n+1}}{1-x}</math>
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|<math>\sum_{i=1}^\infty i x^i = \frac{x}{(1-x)^2}</math>
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|<math>\sum_{i=1}^n i x^i = x\frac{1-x^n}{(1-x)^2} - \frac{n x^{n+1}}{1-x}</math>
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|<math>\sum_{i=1}^{\infty} i^2 x^i =\frac{x(1+x)}{(1-x)^3}</math>
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|<math>\sum_{i=1}^n i^2 x^i =</math> <math>\frac{x(1+x-(n+1)^2x^n+(2n^2+2n-1)x^{n+1}-n^2x^{n+2})}{(1-x)^3} </math>
|-
|<math>\sum_{i=1}^{\infty} i^3 x^i =\frac{x(1+4x+x^2)}{(1-x)^4}</math>
|-
|<math>\sum_{i=1}^{\infty} i^4 x^i =\frac{x(1+x)(1+10x+x^2)}{(1-x)^5}</math>
|-
| <math>\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)</math>  ដែល <math>Li_{s}(x)</math> ជា[[ពហុលោការីត]](Polylogarithm)នៃ <math>x</math>.
|}

=== ភាគបែងធម្មតា ===
* <math>\sum^{\infty}_{i=1} \frac{x^i}i = \log_e\left(\frac{1}{1-x}\right) \!</math> ចំពោះ<math> |x|\le 1, \, x\not= -1</math>

* <math>\sum^{\infty}_{i=0} \frac{(-1)^i}{2i+1} x^{2i+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots = \arctan(x)</math>

* <math>\sum^{\infty}_{i=0} \frac{x^{2i+1}}{2i+1} = \mathrm{arctanh} (x) \!</math>ចំពោះ<math> |x| < 1\!</math>

=== ភាគបែងមាន[[ហ្វាក់តូរ្យែល]](Factorial denominators) ===
ស៊េរីជាច្រើនដែលកើតឡើងពី[[ទ្រឹស្តីបទតាយលើ]](Taylor's theorem)មានមេគុណដែលមានហ្វាក់តូរ្យែល ។

* <math>\sum^{\infty}_{i=0} \frac{x^i}{i!} = e^x</math>

* <math>\sum^{\infty}_{i=0} i   \frac{x^i}{i!} = x e^x</math> (c.f. មធ្យមនៃ[[របាយព័យសុង]](Poisson distribution))
* <math>\sum^{\infty}_{i=0} i^2 \frac{x^i}{i!} = (x + x^2) e^x</math> (c.f. [[ម៉ូម៉ង់ទី២]]នៃរបាយព័យសុង)
* <math>\sum^{\infty}_{i=0} i^3 \frac{x^i}{i!} = (x + 3x^2 + x^3) e^x</math>
* <math>\sum^{\infty}_{i=0} i^4 \frac{x^i}{i!} = (x + 7x^2 + 6x^3 + x^4) e^x</math>


* <math>\sum^{\infty}_{i=0} \frac{(-1)^i}{(2i+1)!} x^{2i+1}=  x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sin x </math>

* <math>\sum^{\infty}_{i=0} \frac{(-1)^i}{(2i)!} x^{2i} =  1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \cos x </math>

* <math>\sum^{\infty}_{i=0} \frac{x^{2i+1}}{(2i+1)!} = \sinh x </math>

* <math>\sum^{\infty}_{i=0} \frac{x^{2i}}{(2i)!} = \cosh x </math>

=== ភាគបែងមានហ្វាក់តូរ្យែលសាំញ៉ាំ(Modified-factorial denominators) ===
* <math>\sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} = \arcsin x\!</math> ចំពោះ<math> |x| < 1\!</math>

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

=== [[ស៊េរីទ្វេធា]](Binomial series) ===
ស៊េរីទ្វេធា (រួមទាំងរឹសការេនៃ <math>\alpha = 1/2</math> ហើយស៊េរីធរណីមាត្រអនន្តចំពោះ <math>\alpha = -1</math>):

[[រឹសការេ]]:
* <math>\sqrt{1+x} = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)n!^24^n}x^n \!</math> ចំពោះ<math> |x|<1\!</math>

[[ស៊េរីធរណីមាត្រ]]:
* <math>(1+x)^{-1} = \sum_{n=0}^\infty (-1)^n x^n \!</math> ចំពោះ<math> |x|<1</math>

ទំរង់ទូទៅ:

* <math>(1+x)^\alpha = \sum_{n=0}^\infty {\alpha \choose n} x^n\!</math> ចំពោះ<math> |x| < 1\!</math> ហើយចំពោះគ្រប់[[ចំនួនកុំផ្លិច]]<math> \alpha\!</math>
:ដោយសំរួលមេគុណទ្វេធា
:: <math>{\alpha\choose n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}\!</math>

* <ref name="ctcs">[http://www.tug.org/texshowcase/cheat.pdf Theoretical computer science cheat sheet]</ref> <math>\sum_{i=0}^\infty {i+n \choose i} x^i = \frac{1}{(1-x)^{n+1}}</math>
* <ref name="ctcs"/> <math>\sum_{i=0}^\infty \frac{1}{i+1}{2i \choose i} x^i = \frac{1}{2x}(\sqrt{1-4x})</math>
* <ref name="ctcs"/> <math>\sum_{i=0}^\infty {2i \choose i} x^i = \frac{1}{\sqrt{1-4x}}</math>
* <ref name="ctcs"/> <math>\sum_{i=0}^\infty {2i + n \choose i} x^i = \frac{1}{\sqrt{1-4x}}\left(\frac{1-\sqrt{1-4x}}{2x}\right)^n</math>

== [[មេគុណទ្វេធា]](Binomial coefficients) ==
* <math>\sum_{i=0}^n {n \choose i} = 2^n</math>
* <math>\sum_{i=0}^n {n \choose i}a^{(n-i)} b^i = (a + b)^n</math>
* <math>\sum_{i=0}^n {i \choose k} = { n+1 \choose k+1 }</math>
* <math>\sum_{i=0}^n {k+i \choose i} = { k + n + 1 \choose n }</math>
* <math>\sum_{i=0}^r {r \choose i}{s \choose n-i} = {r + s \choose n}</math>

== [[អនុគមន៍ត្រីកោណមាត្រ]] ==
ផលបូក នៃ ស៊ីនុស និងកូស៊ីនុស មានក្នុង[[ស៊េរីហ្វូរៀរ]](Fourier series) ។

* <math>\sum_{i=1}^n \sin\left(\frac{i\pi}{n}\right) = 0</math>
* <math>\sum_{i=1}^n \cos\left(\frac{i\pi}{n}\right) = 0</math>

== មិនមានចំណាត់ថ្នាក់ ==
* <math>\sum_{n=b+1}^{\infty} \frac{b}{n^2 - b^2} = \sum_{n=1}^{2b} \frac{1}{2n}</math>

[[ចំនាត់ថ្នាក់ក្រុម:គណិតវិទ្យា]]