What we call a series.—Convergence and divergence.
A geometrical progression is convergent, when the ratio is smaller than unity; diverging, when it is greater.
The terms of a series may decrease indefinitely and the series not be converging.
A series, all the terms of which are positive, is converging, when the ratio of one term to the preceding one tends towards a limit smaller than unity, in proportion as the index of the rank of that term increases indefinitely.—The series is diverging when this limit is greater than unity. There is uncertainty when it is equal to unity.
In general, when the terms of a series decrease indefinitely, and are alternately positive and negative, the series is converging.
Combinations, arrangements, and permutations of m letters, when each combination must not contain the same letter twice.
Development of the entire and positive powers of a binomial.—General terms.
Development of (a + b √-1)m.
Limit towards which (1 + 1/m)m tends, when m increases indefinitely.
Summation of piles of balls.