Homework Help: Series question

1. Apr 21, 2008

daniel_i_l

1. The problem statement, all variables and given/known data
Lets say that I have some sequence $$(a_n)$$ which converges to 0 at infinity and that for all n $$a_{n+1} < a_n$$ but the sequence $$(a_n)$$ diverges. Now I know that the series
$$(cos(n) a_n)$$ converges but can I use the following argument to prove that
$$|cos(n) a_n|$$ doesn't converge:
$$|cos(n) a_n| >= {cos}^{2}(n) a_n = {a_n}/2 + {(cos(2n)) a_n}/2$$
And since $${(cos(2n)) a_n}/2$$ converges and $${a_n}/2$$ diverges
$${cos}^{2}(n) a_n$$ diverges and so $$|cos(n) a_n|$$ diverges.
Is that always true?
Thanks.

2. Apr 21, 2008

Dick

That seems ok, provided you can prove cos(2n)*a_n converges. Are you using Abel-Dedekind-Dirichlet (summation by parts)? I just figured out how it works, so I had to ask.

3. Apr 22, 2008

daniel_i_l

4. Apr 22, 2008

Dick

5. Apr 22, 2008

daniel_i_l

Well I know that cos(n) has bounded partial sums and cos(2n) = 2(cos(n))^2 - 1 so that means that cos(2n) also has bounded partial sums right?

6. Apr 22, 2008

Dick

Not at all! cos(n)^2 doesn't have bounded partial sums and neither does 1! How did you show cos(n) had bounded partial sums?? Try and apply the same technique to cos(2n).

7. Apr 22, 2008

HallsofIvy

You mean series rather than that last "sequence", right?