Consider-the-sequence-defined-by-0-lt-u-0-lt-1-and-n-N-u-n-1-u-n-u-n-2-1-Show-that-the-sequence-u-n-converges-What-is-its-limit-2-Show-that-the-series-with-general-term-u-n-2-conv

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Question Number 127876 by Ar Brandon last updated on 02/Jan/21

$$\mathrm{Consider}\mathrm{the}\mathrm{sequence}\mathrm{defined}\mathrm{by}:0<{\mathrm{u}}_0<1\mathrm{and}\mathrm{\forall }\mathrm{n}\in \mathbb{N},{\mathrm{u}}_{\mathrm{n}+1}={\mathrm{u}}_{\mathrm{n}}-{\mathrm{u}}_{\mathrm{n}}^2.$$$$1.\mathrm{Show}\mathrm{that}\mathrm{the}\mathrm{sequence}\left({\mathrm{u}}_{\mathrm{n}}\right)\mathrm{converges}.\mathrm{What}\mathrm{is}\mathrm{its}\mathrm{limit}?$$$$2.\mathrm{Show}\mathrm{that}\mathrm{the}\mathrm{series}\mathrm{with}\mathrm{general}\mathrm{term}{\mathrm{u}}_{\mathrm{n}}^2\mathrm{converges}.$$$$3.\mathrm{Show}\mathrm{that}\mathrm{the}\mathrm{series}\mathrm{with}\mathrm{general}\mathrm{terms}\ln \left(\frac{{\mathrm{u}}_{\mathrm{n}+1}}{{\mathrm{u}}_{\mathrm{n}}}\right)\mathrm{and}{\mathrm{u}}_{\mathrm{n}}\mathrm{diverge}.$$

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