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Let-u-n-k-1-n-n-n-2-k-2-for-n-N-gt-0-Show-that-u-n-n-gt-0-is-increasing-




Question Number 200521 by brahim_mekkaoui last updated on 19/Nov/23
Let u_n =Σ_(k=1) ^n  (n/(n^2 +k^2 ))  for n∈N_(>0)   .  Show that (u_n )_(n>0)  is increasing.
$$\mathrm{Let}\:{u}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{{n}}{{n}^{\mathrm{2}} +{k}^{\mathrm{2}} }\:\:\mathrm{for}\:{n}\in\mathbb{N}_{>\mathrm{0}} \:\:. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\left({u}_{{n}} \right)_{{n}>\mathrm{0}} \:\mathrm{is}\:\mathrm{increasing}. \\ $$

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