Definition: A sequence (x_n) verconges to x if there exists an > 0 such that for all N N it is true
that n N implies |x_n - x| Give an example of a vercongent sequence. Can you give an example
of a vergonent sequence that is divergent? What exactly is being described in this strange
definition?
Solution
The oscillating sequences are the typical examples of vercongent sequence
Ex = { 1, -1, 1, -1, 1, -1, ... }
The vercongent sequence that is divergent
{5, 2, 5, 2, 5, 2, ....} since the function at n=infinite doesn\'t die down to zero and will oscillate
between two values i,e. 2 and 5
According to the definition of vercongent sequence, the function is said to be constrained on the
difference of the number at n and initial value, but the ratio has not been constrained, if the ratio
was there as in the case of converging sequence then the series will eventually converge
In our case
e- < |2-5| < e+
hence the vlaue of epsilon will be equal to 3 for the above series.
Definition A sequence (x_n) verconges to x if there exists an 0 s.pdf
1. Definition: A sequence (x_n) verconges to x if there exists an > 0 such that for all N N it is true
that n N implies |x_n - x| Give an example of a vercongent sequence. Can you give an example
of a vergonent sequence that is divergent? What exactly is being described in this strange
definition?
Solution
The oscillating sequences are the typical examples of vercongent sequence
Ex = { 1, -1, 1, -1, 1, -1, ... }
The vercongent sequence that is divergent
{5, 2, 5, 2, 5, 2, ....} since the function at n=infinite doesn't die down to zero and will oscillate
between two values i,e. 2 and 5
According to the definition of vercongent sequence, the function is said to be constrained on the
difference of the number at n and initial value, but the ratio has not been constrained, if the ratio
was there as in the case of converging sequence then the series will eventually converge
In our case
e- < |2-5| < e+
hence the vlaue of epsilon will be equal to 3 for the above series
