Lukion taulukot/Summia ja sarjoja

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Summia ja sarjoja

1.	${\begin{aligned}S_{n}&=a+(a+d)+(a+2d)+\ldots +\underbrace {(a+(n-1)d)} _{u}\\&=n\cdot {\frac {a+u}{2}}\end{aligned}}$	aritmeettinen summa $a$ on ensimmäinen ja $u$ viimeinen termi
2.	${\begin{aligned}S_{n}&=a+aq+aq^{2}+\ldots +aq^{n-1}\\&={\frac {a(1-q^{n})}{1-q}}\ (q\neq 1)\end{aligned}}$	geometrinen summa
3.	$a+aq+aq^{2}+\ldots$ $s={\frac {a}{1-q}}$	geometrinen sarja; suppenee, kun $|q|<1$; summa $s$
4.	$1+2+3+\ldots +n={\frac {n(n+1)}{2}}$
5.	$1^{2}+2^{2}+3^{2}+\ldots +n^{2}={\frac {n(n+1)(2n+1)}{6}}$
6.	$1^{3}+2^{3}+3^{3}+\ldots +n^{3}={\frac {n^{2}(n+1)^{2}}{4}}$
7.	${\frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-\ldots \quad (|x|<1)$
8.	$e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\ldots +{\frac {x^{n}}{n!}}+\ldots$