Meaning of an Infinite Series

If \left\langle {{u_n}} \right\rangle be given real valued sequence, then an expression of the form

{u_1} + {u_2} + {u_3} + \cdots + {u_n} + \cdots

is called an infinite series. In symbols it is generally written as \sum\limits_{n = 1}^\infty {{u_n}} or \sum {{u_n}} .

If all the terms of \left\langle {{u_n}} \right\rangle after a certain number are zero then the expression {u_1} + {u_2} + {u_3} + \cdots + {u_m}, written \sum\limits_{n = 1}^m {{u_n}} is called a finite series. Simply speaking, even without any reference to the sequence, an expression of the form {u_1} + {u_2} + {u_3} + \cdots + {u_m} is called a finite series.

Sequence of Partial Sums of an Infinite Series:

An expression of the form {u_1} + {u_2} + {u_3} + \cdots + {u_n} + \cdots 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 {u_1} + {u_2} + {u_3} + \cdots + {u_n} + \cdots a sequence \left\langle {{S_n}} \right\rangle define by, {u_1} + {u_2} + {u_3} + \cdots + {u_n}. The sequence \left\langle {{S_n}} \right\rangle is called the sequence of partial sums of the given series {u_1} + {u_2} + {u_3} + \cdots + {u_n} + \cdots .