prove-that-n-n-n-1-t-dt-2-prove-that-k-1-n-1-k-n-1-1-if-lt-1-and-k-1-n-1-k-ln-n-if-1-

Tinku Tara June 4, 2023 Relation and Functions 0 Comments

Question Number 32369 by prof Abdo imad last updated on 23/Mar/18

prove that n^(−α) ∼ ∫_n ^(n+1) t^(−α) dt 2) prove that Σ_(k=1) ^n (1/k^α ) ∼ (n^(1−α) /(1−α)) if α<1 and Σ_(k=1) ^n (1/k^α ) ∼ ln(n) if α=1 .

$${prove}\:{that}\:\:{n}^{−\alpha} \:\sim\:\int_{{n}} ^{{n}+\mathrm{1}} \:{t}^{−\alpha} {dt} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}^{\alpha} }\:\sim\:\:\frac{{n}^{\mathrm{1}−\alpha} }{\mathrm{1}−\alpha}\:{if}\:\:\alpha<\mathrm{1}\:{and} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}^{\alpha} }\:\sim\:{ln}\left({n}\right)\:{if}\:\alpha=\mathrm{1}\:. \\ $$

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prove-that-n-n-n-1-t-dt-2-prove-that-k-1-n-1-k-n-1-1-if-lt-1-and-k-1-n-1-k-ln-n-if-1-

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