The series ∑ (n=1 to ∞) n^(-1/4) Convergent or Divergent?

Question

I am checking my work and answer. I'm using the Integral test.

∞
∑ n^(-1/4) gives me f(x) = x^(-1/4)
n=1

∞
= ∫ x^(-1/4)dx
1

= lim ∫ x^(-1/4)dx
r->∞

........................ r
= x^(3/4)/(3/4) | = [(r^(3/4))/(3/4)] - [1/(3/4)] = ∞ - (4/3)
....................... 1

Which is Divergent? It doesn't feel right to me.

Answer 1

you did it right, it is divergent. in general, for sums of n^(-p), you need p to be greater than 1 for convergence.

Answer 2

The series does not converge; it goes to infinity.