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The aim of this lecture is to analyze various concepts about sequences: a sequence being bounded, monotone, and convergent. |
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Given a sequence  , one is interested to know: what happens to  as  becomes large. In the example of the area of the unit circle, we expect that for n sufficiently large,  will be a 'good enough' representation of
 , the area of the unit circle, that is, will  come close to the value , the area of the unit circle when n becomes large? How close? Will it come as close as we want? Using the concept of absolute value (which gives the notion of distance on  ) we can express it mathematically as follows: |
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If we consider the geometric visualization of the sequence as a function,  then saying that the number  will be the limit of the sequence means that given any horizontal strip, of width  centered at , all but finitely many of  lie in this strip.
Intuitively, after some stage all the elements of the sequence are close to . Or a 'tail' of the convergent sequence lies inside any small neighborhood of . A sequence which is not convergent is called a divergent sequence. |
such that given any real number 
, we can find a natural number 
such that 
. 
is convergent to 
. To see this, let 
