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 .