a-n-1-a-n-1-n-a-1-1-Assume-we-do-not-know-that-the-general-term-of-this-sequence-is-a-Harmonic-series-even-though-it-is-well-known-as-such-Under-this-assumption-prove-that-the-general-ter

Question Number 228103 by Lara2440 last updated on 22/Mar/26

$${a}_{{n}+\mathrm{1}} −{a}_{{n}} =\frac{\mathrm{1}}{{n}}\:\:,\:{a}_{\mathrm{1}} =\mathrm{1} \\ $$$$\: \\ $$$$\mathrm{Assume}\:\mathrm{we}\:\mathrm{do}\:\mathrm{not}\:\mathrm{know}\:\mathrm{that}\:\mathrm{the}\:\mathrm{general}\:\mathrm{term}\:\mathrm{of}\:\mathrm{this} \\ $$$$\mathrm{sequence}\:\mathrm{is}\:\mathrm{a}\:\mathrm{Harmonic}\:\mathrm{series}\:\mathrm{even}\:\mathrm{though} \\ $$$$\mathrm{it}\:\mathrm{is}\:\mathrm{well}\:\mathrm{known}\:\mathrm{as}\:\mathrm{such}.\:\mathrm{Under}\:\mathrm{this}\:\mathrm{assumption} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{general}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{is}\: \\ $$$$\mathrm{monotonically}\:\mathrm{increasing}. \\ $$

Answered by aleks041103 last updated on 24/Mar/26

$${Wdym}?\:{This}\:{is}\:{obvious}: \\ $$$${a}_{{n}+\mathrm{1}} −{a}_{{n}} =\frac{\mathrm{1}}{{n}}>\mathrm{0}\:\Rightarrow\:{a}_{{n}+\mathrm{1}} >{a}_{{n}} \\ $$

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