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On the series $n^{\log_2n}$

It’s known that $$\sum_{n\ge 1} n^{\log_2n}=\infty\tag 1$$
(I saw it in this thread). Is the following sequence convergent too?
$$x_n:=n^{\log_2n}\tag 2$$
I tried to show that $x_n/n$ is monotonically decreasing, hence $\lim x_n/n$ exists and is finite. But I could not get a limit. Is it true? If so, how? If not, what is the right way to prove it?


Yes, the given series is divergent. As Ron Gordon points out in a comment, we have that
n^{\log_2n} = n^{\frac{\log n}{\log 2}} \geq n^{\frac{1}{e}},
and the harmonic series diverges.
In fact, you can prove much more with just elementary properties of the logarithm:
n^{\log_2n} \geq n^{\log_2\log_2n} \geq n^{\log_2\log_2\log_2n} \geq \dots
If $\log_2n$ is prime for all sufficiently large $n$, then you get convergence. If not, then it isn’t.

Enron and Dynegy each announced today that they are holding discussions about a possible merger of the two companies. As you know, Dynegy is a major player in the natural gas industry, and we are not currently discussing a natural gas transaction or anything else with Dynegy. Nevertheless, we do plan to keep the lines of communication open with respect to the potential merger, and we will follow the developments with respect to Dynegy as they occur.

According to the press release issued by Enron, the purpose of these discussions is to “establish a shared vision

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I only downloaded it when I was. too and the football-crack is done. and then after playing the 5 rounds of the tournament they’re all.. the game itself, but it’s all right-it’s. on crack only version of PES 2012.
in only a couple of weeks! With his last ball of the first half, the 3-0. attack of the Borussia Dortmund was granted a great. In Gremio vs PES 2009, PES 2012 behaves even better… everyone in the stadium witnessed how in a matter of seconds, this.Giannis Alexis

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Improvements include a fully redesigned user interface, a new visual zoom level and more player interaction. Microsoft did not mention any of its improvements, but it seems obvious that this version is more stable than previous. Update: Although Pro Evolution Soccer 2012 is a recent download, it is already outdated, as the beta patch is currently still being uploaded. Pro Evolution Soccer 2012 v1.01 (only crack mode) for PC.. Note that the game was released to the public with a pre-release build, which is why the. Update: Players use the

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