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I would like to show that the following (and other similar formulae) tends to zero.

Limit[Sum[(2*E*n)^w/(w^(n/2+w)), {w,2,n}],n->Infinity]

What's the right way to do this in Mathematica?

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You can play around with ExpToTrig –  Sektor Jan 8 at 20:01
    
If you DiscretePlot your sum you will see that the limit seems to be 0. Proving that (by hand) is rather easy afterwards. –  A.G. Jan 8 at 23:42
    
@A.G. If you change the problem just slightly to Limit[Sum[(2*E*n)^w/(w^(n/4+w)), {w,2,n}],n->Infinity] it is much less obvious how to solve it by plotting. –  marshall Jan 10 at 11:39
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