Need assistance with creating a convergent series whose tail thickness is between the series below.

See picture. Typing out if you cant read picture.

SIGMA (Sqrt n) / (1+n) >> >>SIGMA (Sqrt n) / (1+n^2)

Create a series as before whose tail thickness is between the same series, though this time have that series diverge.

SIGMA (Sqrt n) / (1+n) >> >>SIGMA (Sqrt n) / (1+n^2)