The series \(\sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}\) converges. The comparison test is a suitable method for proving this convergence. By rationalizing the numerator, the series can be rewritten as \(\sum_{n=1}^{\infty}\frac{1}{n(\sqrt{n+1}+\sqrt{n})}\). This transformation allows for a clearer comparison with a known convergent series.
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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.