Question

Using the integral test, how do you show whether `(1/sqrt (n+1))` diverges or converges?

Answer 1

I would assume that this goes from `n = 1` to `oo`? The domain must be `[k,oo]` in order for this to work. If so, I've already answered this here:
http://socratic.org/questions/using-the-integral-test-how-do-you-show-whether-sum-1-sqrt-n-1-diverges-or-conve#151880

Overall, it diverges, because the integral evaluates as `oo`. The gist of it is, integrate it replacing `n` with `x`, and show that it will result in `oo` when evaluated from `1` to `oo`.

You should also know that this is a "p-series":

`sum_(n = 1)^(oo) 1/((npmk)^p)`
where `k in RR` (`k` is in the set of real numbers)

If `p > 1`, the sum will converge, whereas if `p <=1`, the sum will diverge. When `p = 1`, it's called the harmonic series