- #1
Rectifier
Gold Member
- 313
- 4
The problem
In this problem I am supposed to show that the following series converges by somehow comparing it to ## \frac{1}{k\sqrt{k}} ## :
$$ \sum^{\infty}_{k=1} \left( \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} \right) $$
The attempt
## \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} = \frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k}\sqrt{k+1}} ##
Now, I am supposed to somehow find:
## \lim_{k \rightarrow \infty} \frac{\frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k}\sqrt{k+1}}}{\frac{1}{k\sqrt{k}}} ##. I am not sure that this I have simplified the starting expression enough.
There are alternative solutions to this problem but I would like to solve it using limit comparison test. Thank you for your understanding.
In this problem I am supposed to show that the following series converges by somehow comparing it to ## \frac{1}{k\sqrt{k}} ## :
$$ \sum^{\infty}_{k=1} \left( \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} \right) $$
The attempt
## \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} = \frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k}\sqrt{k+1}} ##
Now, I am supposed to somehow find:
## \lim_{k \rightarrow \infty} \frac{\frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k}\sqrt{k+1}}}{\frac{1}{k\sqrt{k}}} ##. I am not sure that this I have simplified the starting expression enough.
There are alternative solutions to this problem but I would like to solve it using limit comparison test. Thank you for your understanding.