Let us consider the sequence $(a_n)_{n \ge 1}$ such that $$a_n=\frac {1}{\sqrt {n^2+1}}+ \frac {1}{\sqrt {n^2+2}} + \dots +\frac {1}{\sqrt {n^2+n}}.$$ Show that the sequence is not monotone.
I found the problem, but the solution is wrong.
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1This sequence is an increasing one. It converges to $1$. – Adren Feb 15 '17 at 19:45
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Also: http://math.stackexchange.com/questions/948049/how-to-prove-that-lim-n-to-infty-left-sum-r-1n-dfrac-1-sqrtn2-r. – Martin R Feb 15 '17 at 20:05