Monday, 3 March 2014

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 ku is convergent, divergent or oscillatory 

according as , summation of  u 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 .

No comments:

Post a Comment

Our Latest Post

How to find log (alpha+ i beta), Where alpha and beta are real

Here is the video to show the details of solving this problem. It is an important problem for basic understanding about the logarithm of re...

Popular Post