series

Series

For numbers $a_1,a_2,a_3,\dots$ we define:

$$\sum _{i=1}^{\infty }{a}_i$$

as the limit of the partial sums:

$$\underset{n\to \infty }{\lim }{S}_n=\underset{n\to \infty }{\lim }\sum _{i=1}^n{a}_i$$

If this limit approaches a finite number $L$, we say the series converges to $L$:
If it blows up to $\infty$ or oscillates forever, we say it diverges.

EXAMPLE

$1/2+1/4+1/8+1/16+\cdots \to 1$ converges.
$1+1+1+1+\cdots \to \infty$ diverges.

Note

For nonnegative terms $a_i\ge 0$, the partial sums $S_n$ are monotonically increasing, so the limit always exists in $[0,+\infty ]$.