# Series - Converge or diverge

• Jun 11th 2010, 01:54 AM
4Math
Series - Converge or diverge
Hello,

How can I show if these series are convergent or divergent?

k = 1∞ ((k!)^(1/k))/(k)

For the following series:

k = 1∞ (e^(3ik))/(k^(3/2))

I have started a solution here:http://img819.imageshack.us/img819/6864/solution.jpg
and was wondering if it is correct. Why is (e^(3ik))=1 when k->∞

Appriciate any guidance and a detail explanation.

Thank you
• Jun 11th 2010, 02:02 AM
$\displaystyle |e^{3ik}|= 1$ because it's magnitude is 1.
• Jun 11th 2010, 02:14 AM
simplependulum
Quote:

Originally Posted by 4Math
Hello,

How can I show if these series are convergent or divergent?

k = 1∞ ((k!)^(1/k))/(k)

For the following series:

k = 1∞ (e^(3ik))/(k^(3/2))

I have started a solution here:http://img819.imageshack.us/img819/6864/solution.jpg
and was wondering if it is correct. Why is (e^(3ik))=1 when k->∞

Appriciate any guidance and a detail explanation.

Thank you

$\displaystyle \sqrt[k]{k!} > 1$ so

$\displaystyle \frac{\sqrt[k]{k!} }{k} > \frac{1}{k} > 0$
• Aug 6th 2010, 11:48 AM
4Math
Quote:

Originally Posted by simplependulum
$\displaystyle \sqrt[k]{k!} > 1$ so

$\displaystyle \frac{\sqrt[k]{k!} }{k} > \frac{1}{k} > 0$

Is this mean that the seies convergent or divergent??
• Aug 6th 2010, 01:25 PM
Plato
Does $\displaystyle \displaystyle \sum\limits_k {\frac{1}{k}}$ converge or diverge?