Proving the Cauchy Sequence of (An)

[hypermonkey2]

How could i show that the sequence
(An)= (1+(1/sqrt(2))+(1/Sqrt(3))+...+(1/sqrt(n))-2sqrt(n))) is Cauchy?
Thanks in advance!

 

[Kummer]

How could i show that the sequence
(An)= (1+(1/sqrt(2))+(1/Sqrt(3))+...+(1/sqrt(n))-2sqrt(n))) is Cauchy?
Thanks in advance!

You need to show it is convergent.
Define f(x) = \frac{1}{\sqrt{x}}. Now confirm that \sum_{j=1}^{n} \frac{1}{\sqrt{j}} \geq \int_1^n \frac{1}{\sqrt{x}}dx. Use this to create a lower bound. Now apply the monotone theorem.

 

[hypermonkey2]

Hmm.. I was hoping that there would be a way to do this straight from the definition of a Cauchy sequence, without use of the notion of a definite integral.. thanks though! any other ideas?