# Thread: What Test Do I Use and How?

1. ## What Test Do I Use and How?

Could someone explain how to use what tests to determine whether the sum from n=1 to infinity of (cos(n+9)/(n^(3/2))) is absolutely convergent, conditionally convergent, or divergent? Could I use the direct comparison or limit comparison for this? If so, how; what would I compare it to?

2. ## Re: What Test Do I Use and How?

Since $\displaystyle \left| \cos{(x)} \right| \leq 1$ for all x, that means your absolute value series will be less than the series of $\displaystyle \frac{1}{n^{\frac{3}{2}}}$.

This is a convergent p-series, so by the comparison test, your series is absolutely convergent.

3. ## Re: What Test Do I Use and How?

Originally Posted by Prove It
Since $\displaystyle \left| \cos{(x)} \right| \leq 1$ for all x, that means your absolute value series will be less than the series of $\displaystyle \frac{1}{n^{\frac{3}{2}}}$.

This is a convergent p-series, so by the comparison test, your series is absolutely convergent.
So, if I were to use the limit comparison test, would I come to the same conclusion? Is it true that if b(sub n) converges and the limit as n goes to infinity of (a(sub n)/b(sub n)) approaches infinity, then a(sub n) also converges?

4. ## Re: What Test Do I Use and How?

That's exactly what I did use...

5. ## Re: What Test Do I Use and How?

Originally Posted by Preston019
So, if I were to use the limit comparison test, would I come to the same conclusion? Is it true that if b(sub n) converges and the limit as n goes to infinity of (a(sub n)/b(sub n)) approaches infinity, then a(sub n) also converges?
There is no need for any further tests.
$\sum\limits_{n = 1}^\infty {\left| {\frac{{\cos (n)}}{{\sqrt {{n^3}} }}} \right|} \le \sum\limits_{n = 1}^\infty {\frac{1}{{\sqrt {{n^3}} }}}$ tells you that your series converges absolutely.

If a series converges absolutely it converges period.

6. ## Re: What Test Do I Use and How?

Originally Posted by Prove It
That's exactly what I did use...
You just said comparison test, you didn't specify whether you used the direct comparison test or the limit comparison test.

Would my second question be true then?

7. ## Re: What Test Do I Use and How?

Originally Posted by Plato
There is no need for any further tests.
$\sum\limits_{n = 1}^\infty {\left| {\frac{{\cos (n)}}{{\sqrt {{n^3}} }}} \right|} \le \sum\limits_{n = 1}^\infty {\frac{1}{{\sqrt {{n^3}} }}}$ tells you that your series converges absolutely.

If a series converges absolutely it converges period.
I know, I'm just trying to figure out whether
...if b(sub n) converges and the limit as n goes to infinity of (a(sub n)/b(sub n)) approaches infinity, then a(sub n) also converges?
is true.