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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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