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Need assistance with creating a convergent series whose tail thickness is between the series below. See picture. Typing out if you cant read picture....

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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