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


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

$p r o v e t h a t n^{- α} \sim \int_{n}^{n + 1} t^{- α} d t$ $2) p r o v e t h a t \sum_{k = 1}^{n} \frac{1}{k^{α}} \sim \frac{n^{1 - α}}{1 - α} i f α < 1 a n d$ $\sum_{k = 1}^{n} \frac{1}{k^{α}} \sim l n (n) i f α = 1 .$
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