Sequence of Partial Sums of a Series

 


Let;   u1+u2+ ..............+un+.......

be an infinite series . 

If  Sn denotes the sums of first   n   terms of the series ,

So that ;

Sn=  u1+u2+ ..............+un+,   then ; the sequence   {Sn}   is called the sequence of partial sums of the given series .

Convergent Series :-   If    {Sn}  tends to a finite and infinite limit  S,  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. 

Thus   S   is defined as

lim n-->infinity  Sn=S

or,  Sn-->S,  as  n-->infinity 

Divergent Series :-  If  {Sn}  tends to +infinity  or  -infinity

as  n tends to  infinity , then the series is said to be divergent .

in other words , the series is divergent if having given any positive number delta whatsoever , we can find a finite number   m  such that   Sn>delta  ,
when   n is less and equal to m .

Oscillatory Series :-  If  {Sn} tends to no definite limit whether  finite or infinite as   n tends to infinity   , then the series is aid to be Oscillate. We say that the series oscillates finitely or infinitely according as  Sn  oscillates between finite limits or between positive and negative infinity .