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let-u-n-k-1-n-n-k-2-n-2-for-n-N-gt-0-show-that-u-n-n-N-gt-0-is-increasing-




Question Number 200759 by brahim_mekkaoui last updated on 23/Nov/23
let u_n =Σ_(k=1) ^n (n/(k^2 +n^2 )) for n∈N_(>0)     show that (u_n )_(n∈N_(>0) ) is increasing.
$$\mathrm{let}\:\mathrm{u}_{\mathrm{n}} =\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{n}}{\mathrm{k}^{\mathrm{2}} +\mathrm{n}^{\mathrm{2}} }\:\mathrm{for}\:\mathrm{n}\in\mathbb{N}_{>\mathrm{0}} \:\: \\ $$$$\mathrm{show}\:\mathrm{that}\:\left(\mathrm{u}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}_{>\mathrm{0}} } \mathrm{is}\:\mathrm{increasing}. \\ $$

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