Upper Limit

Let the greatest term $H$ of a Sequence be a term which is greater than all but a finite number of the terms which are equal to $H$. Then $H$ is called the upper limit of the Sequence.

An upper limit of a Series

$$\mathop{\rm upper} \lim_{n\to \infty} S_n = \overline{\lim_{n\to \infty}} S_n=k$$

is said to exist if, for every $\epsilon>0$, $\vert S_n-k\vert < \epsilon$ for infinitely many values of $n$ and if no number larger than $k$ has this property.

See also Limit, Lower Limit

References

Bromwich, T. J. I'a and MacRobert, T. M. ``Upper and Lower Limits of a Sequence.'' §5.1 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 40, 1991.