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Summaus

Kerrataan summausta ja summausmerkintää. Jos $a_1,a_2,\ldots,a_n$ ovat reaalilukuja, niin merkitään

$$\sum_{i=1}^na_i=a_1+a_2+\cdots+a_n.$$

Esimerkiksi

$$\sum_{i=1}^{5}i^2=1^2+2^2+3^2+4^2+5^2=1+4+9+16+25=55.$$

Summausindeksin nimi voidaan valita vapaasti, joskin yleensä käytetään kirjainta $i$, $j$, $k$, $l$, $m$ tai $n$. Indeksointi voidaan aloittaa muustakin indeksistä kuin $1$. Esimerkiksi edellinen summa voidaan kirjoittaa

$$\sum_{i=1}^5i^2=\sum_{k=1}^5k^2=\sum_{j=2}^6(j-1)^2=\sum_{j=0}^4(j+1)^2.$$

Jos termeillä on yhteinen tekijä $c$, niin voidaan laskea

$$\sum_{i=1}^nca_i =(ca_1)+(ca_2)+\cdots+(ca_n) =c(a_1+a_2+\cdots+a_n) =c\sum_{i=1}^na_i$$

eli

$$\sum_{i=1}^nca_i=c\sum_{i=1}^na_i.$$

Samaan tapaan saadaan

$$\sum_{i=1}^n(a_i+b_i)=\sum_{i=1}^na_i+\sum_{i=1}^nb_i.$$

Esimerkiksi

$$\sum_{i=1}^5(7i^2-4i) =7\sum_{i=1}^5i^2-4\sum_{i=1}^5i =7\cdot55-4\cdot15=325.$$

Tärkeä erikoistapaus on vakiotermin $c$ summa

$$\sum_{i=1}^nc=\underbrace{c+c+\cdots+c}_{n\text{ kappaletta}}=nc.$$

Erityisesti

$$\sum_{i=1}^n1=n.$$

Merkin vaihtelu saadaan aikaan luvun $-1$ potensseilla, sillä

$$\begin{split}(-1)^i = \begin{cases} -1,&\text{kun } i \text{ on pariton}\\ 1,&\text{kun } i \text{ on parillinen.} \end{cases}\end{split}$$

Esimerkiksi

$$\sum_{i=1}^5(-1)^ii=-1+2-3+4-5=-3$$

ja

$$\sum_{i=1}^5\frac{(-1)^{i+1}}{i^2}=1-\frac14+\frac19-\frac{1}{16}+\frac{1}{25}=\frac{821}{979}.$$

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