define sequence of real numbers and its convergence with example.
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One of the two most important ideas in Real analysis is that of convergence of a sequence.
Definition
A sequence in R is a list or ordered set: (a1 , a2 , a3 , ... ) of real numbers.
Examples
One may define a sequence (an) by giving an explicit formula for the nth term.
(1/n) = ( 1 , 1/2 , 1/3 , ... )
(sin(πn/4)) = (1/√2 ,1 , 1/√2 , 0 , -1/√2 ,-1 , -1/√2 , 0 , ... )
( (5n+2)/(3n+1) ) = ( 7/4 , 12/7 , 17/10 , 22/13 , 27/16 , ... )
Definition
A sequence in R is a list or ordered set: (a1 , a2 , a3 , ... ) of real numbers.
Examples
One may define a sequence (an) by giving an explicit formula for the nth term.
(1/n) = ( 1 , 1/2 , 1/3 , ... )
(sin(πn/4)) = (1/√2 ,1 , 1/√2 , 0 , -1/√2 ,-1 , -1/√2 , 0 , ... )
( (5n+2)/(3n+1) ) = ( 7/4 , 12/7 , 17/10 , 22/13 , 27/16 , ... )
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