Direct Consequences of Convergence of Series

 

Followings are the Direct Consequences of  the Definition of Convergence of Series :-

1)  -----  The alteration , addition or omission of a finite number of terms of a series has no effect on its being Convergent , Divergent or Oscillatory .

2)  -----  Multiplication of all the terms of a series by a fixed non zero number has no effect on its vbeing convergent , divergent or oscillatory

i.e.   summation of kun  is convergent, divergent or oscillatory 

according as , summation of  un  is convergent, divergent or oscillatory .


It is important here that the sum of infinity of a convergent series is essentially a limit and is not a sum in the sense described in the defination of addition . We are therefore not justified in assuming that the sum of an infinite series is unaltered by changing the order of the terms or by the introduction or removal of brackets .

In fact, changes of this kind may alter the sum or they may transform a convergent series into one which diverges or oscillates .

For example , the series 

(1-1)+(1-1)+(1-1)+...................... is convergent and its sum is zero , but the series 

1-1+1-1+.............. which is obtained by removing the brackets of the original series , oscillates .