Help. I need to test for convergence for the following, but I'm not sure how to do them.?

Question

Sum (sqrt(k+1) - sqrt(k))/sqrt(k)

sum of k!/k^k

Answer 1

and the 2nd uses the Ratio Test, converges ...1/e

for the 1st I would let u² = x in the Integral Test---> {2 √(u² +1) - 2u} du and an argument that this is not really " ∞ - ∞" but more like 0 + ∞

Answer 2

for the first one, you can use integral test.

the integral yields an inverse hyperbolic sine (or the natural logarithm function) .. . . and it would be evaluated up to infinity

which would then make the integral divergent.

thus the series is divergent.