# 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 ${u_1} + {u_2} + {u_3} + \cdots + {u_n} + \cdots$.