daniel_i_l

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**1. Homework Statement**

Lets say that I have some sequence [tex](a_n)[/tex] which converges to 0 at infinity and that for all n [tex]a_{n+1} < a_n [/tex] but the sequence [tex](a_n)[/tex] diverges. Now I know that the series

[tex](cos(n) a_n) [/tex] converges but can I use the following argument to prove that

[tex]|cos(n) a_n| [/tex] doesn't converge:

[tex] |cos(n) a_n| >= {cos}^{2}(n) a_n = {a_n}/2 + {(cos(2n)) a_n}/2 [/tex]

And since [tex]{(cos(2n)) a_n}/2 [/tex] converges and [tex] {a_n}/2 [/tex] diverges

[tex] {cos}^{2}(n) a_n [/tex] diverges and so [tex] |cos(n) a_n| [/tex] diverges.

Is that always true?

Thanks.