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lets-x-gt-0-and-take-the-sequence-a-a-0-x-a-n-1-x-a-n-i-proof-that-0-a-n-a-n-1-ii-proof-that-M-such-that-a-n-M-iii-using-i-and-ii-proof-that-lim-n-a-n-exist-iv-compute-lim-n




Question Number 1635 by 123456 last updated on 28/Aug/15
lets x>0, and take the sequence a  a_0 =(√x)  a_(n+1) =(√(x+a_n ))  i.proof that 0≤a_n ≤a_(n+1)   ii.proof that ∃M such that a_n ≤M  iii.using i and ii proof that lim_(n→∞)  a_n  exist  iv.compute lim_(n→∞)  a_n
$$\mathrm{lets}\:{x}>\mathrm{0},\:\mathrm{and}\:\mathrm{take}\:\mathrm{the}\:\mathrm{sequence}\:{a} \\ $$$${a}_{\mathrm{0}} =\sqrt{{x}} \\ $$$${a}_{{n}+\mathrm{1}} =\sqrt{{x}+{a}_{{n}} } \\ $$$$\mathrm{i}.\mathrm{proof}\:\mathrm{that}\:\mathrm{0}\leqslant{a}_{{n}} \leqslant{a}_{{n}+\mathrm{1}} \\ $$$$\mathrm{ii}.\mathrm{proof}\:\mathrm{that}\:\exists\mathrm{M}\:\mathrm{such}\:\mathrm{that}\:{a}_{{n}} \leqslant\mathrm{M} \\ $$$$\mathrm{iii}.\mathrm{using}\:\mathrm{i}\:\mathrm{and}\:\mathrm{ii}\:\mathrm{proof}\:\mathrm{that}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} \:\mathrm{exist} \\ $$$$\mathrm{iv}.\mathrm{compute}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} \\ $$

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