The discussion revolves around proving the convergence of the series \(\sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}\). Participants are exploring methods to establish convergence, particularly through comparison tests.
The discussion is ongoing, with participants sharing their attempts and seeking further guidance on comparison series. There is no explicit consensus yet, but some productive lines of reasoning are being explored.
Participants express difficulty in identifying appropriate series for comparison, indicating a potential gap in foundational knowledge or assumptions regarding series behavior.
Try rationalizing the numerator.analysis001 said:Homework Statement
Prove that the series \sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n} converges.
The Attempt at a Solution
I think I'm going to use the comparison test but I'm having trouble coming up with a series to compare it to. Any clues would be great. Thanks!
SammyS said:Try rationalizing the numerator.
Let's see ...analysis001 said:Yeah, I've gotten to that point, so as of now I have: \sum_{n=1}^{\infty}\frac{1}{n(\sqrt{n+1}+\sqrt{n})} but I'm still not sure what to compare it to.