Need Assistance With Creating A Convergent Series Whose Tail Thickness Is Betwee

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)  

 

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