find-lim-n-1-n-1-1-n-2-1-n-p-pfixed-fromN-

Question Number 30758 by abdo imad last updated on 25/Feb/18

f i n d {l i m}_{n \to \infty} (\frac{1}{n + 1} + \frac{1}{n + 2} + \dots . + \frac{1}{n + p}) p f i x e d {f r o m N}^{★}

Commented by abdo imad last updated on 28/Feb/18

l e t p u t u_{n} = \frac{1}{n + 1} + \frac{1}{n + 2} + \dots + \frac{1}{n + p}

u_{n} = H_{n + p} - H_{n} = l n (n + p) + γ + o (\frac{1}{n}) - l n (n) - γ - o^{^{'}} (\frac{1}{n})

= l n (\frac{n + p}{n}) + o {(\frac{1}{n})}_{n \to \infty} \to 0 \Rightarrow {l i m}_{n \to \infty} u_{n} = 0