# series convergent or divergent?

• Nov 8th 2010, 09:57 AM
sdh2106
series convergent or divergent?
Could someone help me show the convergence/divergenceof the following (without the Dirichlet test or Abel sum test)?

sum(cos(n)/n) from n=1 to inf

I cannot use the comparison test since cos(n) alternates and our prof. does not want us to use Dirichlet test. Could someone please help?

Thanks!
• Nov 8th 2010, 10:03 AM
Drexel28
Quote:

Originally Posted by sdh2106
Could someone help me show the convergence/divergenceof the following (without the Dirichlet test or Abel sum test)?

sum(cos(n)/n) from n=1 to inf

I cannot use the comparison test since cos(n) alternates and our prof. does not want us to use Dirichlet test. Could someone please help?

Thanks!

If you're doing this rigorously I would note that if $\displaystyle \displaystyle s_m=\sum_{n=1}^{m}\frac{\cos(n)}{n}$ then $\displaystyle \displaystyle s_m=\text{Re}\sum_{n=1}^{m}\frac{e^{in}}{n}$ and go from there.
• Nov 8th 2010, 10:35 AM
Krizalid
the op asked for convergence proof, not for sum value. :D