Proving That The Series is Convergent or Divergent

[jsewell94]

Homework Statement

Determine whether the following series converges or diverges:
\sum_{}^{} ( \frac{1}{3} )^{ln(n)}

Homework Equations

N/A

The Attempt at a Solution

See attached document..

I had my Calc 2 final today, and this was our hard problem...but I don't know if my method is valid or not. Could you help me determine if it is? And if it's not, could you tell me where the flaw in my reasoning is/a better method?

 

[Dick]

Homework Statement

Determine whether the following series converges or diverges:
\sum_{}^{} ( \frac{1}{3} )^{ln(n)}

Homework Equations

N/A

The Attempt at a Solution

See attached document..

I had my Calc 2 final today, and this was our hard problem...but I don't know if my method is valid or not. Could you help me determine if it is? And if it's not, could you tell me where the flaw in my reasoning is/a better method?

It looks ok to me. You could shorten the whole argument up a lot. 3^{ln(n)}=(e^{ln(3)})^{ln(n)}=e^{ln(3) ln(n)}=(e^{ln(n)})^{ln(3)}=n^{ln(3)}. Just use ln(3) for k.

 

[jsewell94]

It looks ok to me. You could shorten the whole argument up a lot. 3^{ln(n)}=(e^{ln(3)})^{ln(n)}=e^{ln(3) ln(n)}=(e^{ln(n)})^{ln(3)}=n^{ln(3)}. Just use ln(3) for k.

Oh wow, that makes sense!

Yeah, I just..couldn't figure out the easy way in the 15 minutes I had left on the test :(
Yay! That means I was the only one who got it :D