Suppeneva sarja

Jos ${\displaystyle \sum _{k=1}^{\infty }x_{k}}$ suppenee, niin ${\displaystyle \lim \limits _{k\to \infty }x_{k}=0}$

Suppenevalle sarjalle erotusta

${\displaystyle R_{n}=S-S_{n}}$

sanotaan sarjan n:nneksi jäännöstermiksi.

Suppenevalle sarjalle ${\displaystyle \lim \limits _{n\to \infty }R_{n}=0}$

Jos ${\displaystyle \sum _{k=1}^{\infty }x_{k}=X}$ ja ${\displaystyle \sum _{k=1}^{\infty }y_{k}=Y}$, sekä ${\displaystyle a\in \mathbb {R} }$, niin

${\displaystyle a)\sum _{k=1}^{\infty }(x_{k}+y_{k})=X+Y}$

${\displaystyle b)\sum _{k=1}^{\infty }ax_{k}=aX}$

Jos sarja ${\displaystyle \sum _{k=1}^{\infty }x_{k}}$ suppenee ja sarja ${\displaystyle \sum _{k=1}^{\infty }y_{k}}$ hajaantuu, niin summasarja ${\displaystyle \sum _{k=1}^{\infty }(x_{k}+y_{k})}$ hajaantuu. Jos molemmat sarjat ${\displaystyle \sum _{k=1}^{\infty }x_{k}}$ ja ${\displaystyle \sum _{k=1}^{\infty }y_{k}}$ hajaantuvat, niin niiden summasarja ${\displaystyle \sum _{k=1}^{\infty }(x_{k}+y_{k})}$ voi joko a)supeta tai b)hajaantua.

Sarja ${\displaystyle \sum _{k=1}^{\infty }x_{k}}$ suppenee ${\displaystyle \Longleftrightarrow \varepsilon >0}$ kohti ${\displaystyle \exists n_{\varepsilon >0}\in \mathbb {N} }$ s.e.

${\displaystyle \vert x_{n+1}+x_{n+2}+x_{n+3}+...+x_{n+p}\vert <\varepsilon }$

kaikilla ${\displaystyle p\in \mathbb {N} }$ aina kun ${\displaystyle n>n_{\varepsilon }}$

sarja ${\displaystyle \sum _{k=1}^{\infty }x_{k}}$ suppenee itseisesti, jos sarja ${\displaystyle \sum _{k=1}^{\infty }\vert x_{k}\vert }$ suppenee.

Jos ${\displaystyle \sum _{k=1}^{\infty }\vert x_{k}\vert }$ suppenee, niin ${\displaystyle \sum _{k=1}^{\infty }x_{k}}$ suppenee. Tällöin sarjoille pätee

${\displaystyle \vert \sum _{k=1}^{\infty }x_{k}\vert \leqslant \sum _{k=1}^{\infty }|x_{k}|}$