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then  ,
 .
Hence,  .
Click here to View the Interactive animation : Applet 1.6 |
(ii)
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The sequence  is not convergent.
Suppose,  . Then given , there exists  such that
 ,
i.e.,
 ,
which is not true (Archimedian property). In some sense is not convergent as it outgrows every real number.
In fact, if a sequence is convergent, it can not grow arbitrarily, as we shall see in the next section. |
(iii)
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Consider the sequence  . Every odd term of the sequence is  and every even term of the sequence is  . Intuitively, the elements  do not come closer to a single value  . We expect to be divergent. We can write it as follows. First suppose  . If  then given  , there should exist some  such that |
.
. Now, if we choose 
(which is possible by the Archimedian property), 
