### 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

*is convergent, divergent or oscillatory*

**kun**according as , summation of

*is convergent, divergent or oscillatory .*

**un**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 .

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