If is 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 the 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 defined by, . The sequence is called the sequence of partial sums of the given series .