contestada

Which of the following statements do not necessarily imply that (an) is divergent? (A) (an) is eventually positive and (1/an) is null (B) (an) is unbounded (C) (an) has two convergent subsequences whose limits are not equal (D) (an) has both an increasing subsequence and a decreasing subsequence (E) All statements imply that (an) is divergent

Respuesta :

josebarradas1607

Answer:

D does not implies that the sequence is divergent. All others statements do.

Step-by-step explanation:

Statement D: "[tex](a_n)[/tex] has both an increasing subsequence and a decreasing subsequence" does not necessarily implies that the sequence is divergent. For example, let [tex](a_n)[/tex] the sequence given by:

[tex]a_n=\frac{1}{n}[/tex] if n is odd

[tex]a_n=-\frac{1}{n}[/tex] if n is even

We can see that the subsequence [tex]a_{2n-1}[/tex] is a decreasing sequence   (the subsequence given by odd indexes). And the subsequence [tex]a_{2n}[/tex] is an increasing sequence   (the subsequence given by even indexes).

However, [tex](a_n)[/tex] is a convergent sequence with limit zero.

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