Meaning of an Infinite Series

If $$\left\langle {{u_n}} \right\rangle $$ is 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 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 $${u_1} + {u_2} + {u_3} + \cdots + {u_n} + \cdots $$ a sequence $$\left\langle {{S_n}} \right\rangle $$ defined 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 $${u_1} + {u_2} + {u_3} + \cdots + {u_n} + \cdots $$.