Let;

*..*

**u1+u2+ ..............+un+.....**be an infinite series .

If

*denotes the sums of first*

**Sn***terms of the series ,*

**n**So that ;

*, then ; the sequence*

**Sn= u1+u2+ ..............+un+***is called the sequence of partial sums of the given series .*

**{Sn}****Convergent Series :-**If

*tends to a finite and infinite limit*

**{Sn}***then*

**S,***is defined to be the sums to infinity of the series and the series is said to be convergent to the sum*

**S**

**S.**Thus

*is defined as*

**S**

**lim n-->infinity Sn=S**or,

**Sn-->S, as n-->infinity****Divergent Series :-**If

*tends to*

**{Sn}**

**+infinity or -infinity**as

*, then the series is said to be divergent .*

**n tends to infinity**in other words , the series is divergent if having given any positive number delta whatsoever , we can find a finite number

*such that*

**m**

**Sn>delta ,**when

**n is less and equal to m .****Oscillatory Series :-**If

*tends to no definite limit whether finite or infinite as*

**{Sn}***, then the series is aid to be Oscillate. We say that the series oscillates finitely or infinitely according as*

**n****tends to infinity***oscillates between finite limits or between*

**Sn***.*

**positive and negative infinity**