Hello folkx :) Wolfram|Alpha is RIGHT, this series converges. Here is the proof by a friend of mine (Jean-Pierre Delgado) :

Well it is in French but so simple to read :) Have fun and keep cool :) Jean-Michel Collard

Hello Shengui and others, Yes the proof USING Mathematica is clear enough and is absolutely right. But what I expected is a formal proof of the divergence withOUT using Mathematica. I just need a clue to start to prove it with pen and paper. Thank you for the very high quality of your post!

Jean-Michel Collard

Looks like performance issue.

Probably at the beginning Wolfram|Alpha looks for analytical solution if not found then use a numerically approximation.

To quick calculation on web page Wolfram|Alpha use 200 terms but not very big number.

Probably,if the change value of the sum is small, it concludes that the sum is convergent.

If 200 is a close approximation to $\infty$, then Sum[k^(Cos[1/k] - 2), {k, 1, 200}] // N = 5.78299.

But if one plots the partial sum vs the number of terms, one sees the following:

t = Table[{n, Sum[k^(Cos[1/k] - 2), {k, 1, n}]}, {n, 1, 5000, 100}];
ListLogLinearPlot[Table[{n, Sum[k^(Cos[1/k] - 2), {k, 1, n}]}, {n, 1, 5000, 100}], PlotRange -> All]

So I think you're correct in that the sum does not converge. But I don't know why Wolfram Alpha gives the answer it does.