Find limit of the sequence$$a_n = \frac{1}{n}\sum_{1\leq i<j\le n}\frac{1}{\sqrt[]{ij}}.$$
I have no idea how to solve it. I don't know where to start.
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Anne Bauval
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LSSPLY
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1your question is not clear, retype your problem in the question text space and not just on the title – hellofriends Oct 17 '23 at 17:49
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1Quick beginner guide for asking a well-received question + please avoid "no clue" questions: Too many questions begin or end with "I don't even know how to begin with this problem". While this may be true (you may genuinely have no idea how to approach the problem), it is still not a valid reason to limit your post to the statement of the problem without any mention of your own thoughts. – Anne Bauval Oct 17 '23 at 17:55
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Hint: use $\lim_{j\to\infty}\frac{1}{\sqrt j}\sum_{i=1}^j\frac{1}{\sqrt i}=2$ plus Cesaro theorem. – Anne Bauval Oct 17 '23 at 18:14
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can you please provide one more hint. I don't know how to use this limit in $a_n$ – LSSPLY Oct 17 '23 at 19:05
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The sequence $$b_j:=\frac1{\sqrt j}\sum_{1\le i<j}\frac1{\sqrt i}$$ converges to $2$, hence so does its Cesàro mean $$a_n=\frac1n\sum_{j=1}^nb_j.$$
Anne Bauval
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