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-

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 .