squeeze theorem: Sn=1/sqrt(n^2+1)+1/sqrt(n^2+2)+...+1/sqrt(

[shakalandro]

If Sn = 1/sqrt(n^2 + 1) + 1/sqrt(n^2 + 2) + ... + 1/sqrt(n^2 + n), show that (1 + 1/n)^-.5 < Sn < 1. I cannot find a way to show this and I need help.

 

[DrMike]

If Sn = 1/sqrt(n^2 + 1) + 1/sqrt(n^2 + 2) + ... + 1/sqrt(n^2 + n), show that (1 + 1/n)^-.5 < Sn < 1. I cannot find a way to show this and I need help.

What happens if you replaces all the terms in Sn by the biggest term? You'll get something bigger than Sn, but what?

And what happens if you replace all the terms by the smallest term?

Now you have some bounds on Sn. What are they? Do they imply (1 + 1/n)^-.5 < Sn < 1 ?

I haven't tried doing this, but it is the first thing I would try.