Thread: Series - Converge or diverge

1. 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

2. $|e^{3ik}|= 1$ because it's magnitude is 1.

3. 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

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

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

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

$\frac{\sqrt[k]{k!} }{k} > \frac{1}{k} > 0$
Is this mean that the seies convergent or divergent??

5. Does $\displaystyle \sum\limits_k {\frac{1}{k}}$ converge or diverge?