Sequence and Series

A set of numbers arranged in a definite order according to some definite rule is called a sequence. A sequence is a function whose domain is the set N of natural numbers.

Indicated sum of the terms in a sequence is called a series. The result of performing the additions is the sum of the series.


General Properties of Sequence and Series in Mathematics

Properties of sequence and series in mathematics:

1) The convergence or divergence of an infinite series remains unaffected by the addition or removal of finite numbers.

2) If a series in which all terms are positive is convergent, the series remains convergent even when some or all its terms are negative

3) The convergence or divergence of an infinite series remains unaffected by multiplying each term by a finite number.

Some Facts of Sequence and Series in Mathematics

Convergence, Divergence and Oscillation of a series:

Consider the infinite series Σun=u1+u2+.....+un+......∞

and let the sum of the first n terms be Sn=u1+u2+u3+.....+un

Clearly Sn is a function of n and as increases indefinitely three possibilities arise:

1)If Sn tends to finite limit as n→∞,Σun is convergent

2)If Sntends to ±∞ asn→∞,then Σun is divergent

3)If Sndoes not tend to unique limit as n→∞ then Σunis oscillatory.