If be given real valued sequence, then an expression of the form
is called an infinite series. In symbols it is generally written as or .
If all the terms of after a certain number are zero then the expression , written is called a finite series. Simply speaking, even without any reference to the sequence, an expression of the form is called a finite series.
Sequence of Partial Sums of an Infinite Series:
An expression of the form which involves addition of infinitely many terms has in itself no meaning. In order to give a meaning to the value of such an infinite sum, we form a sequence of partial sums. It is the limit of such a sequence which gives meaning to the infinite series.
Let us associate to the infinite series a sequence define by, . The sequence is called the sequence of partial sums of the given series .