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.

Thank you Jim and Shenghui! Then we agree on that the sum in fact is divergent! Shenghui, thanks for reporting the issue to the Wolfram|Alpha math team. Could you please let me know if/when they find out what went wrong and how they were able to fix the issue? Thanks in advance!

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