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Question-210754




Question Number 210754 by mokys last updated on 18/Aug/24
Answered by Berbere last updated on 19/Aug/24
1+N^r ∼N^r   lim_(n→∞) ((Σ_(k=1) ^N k^s )/N^s )=lim_(N→∞) Σ_(k=1) ^N (1/N)((k/N))^s =∫_0 ^1 x^s =(1/(s+1))=((r−s)/r)  1=r−s  s+1=r  lim_(N→∞) ((Σ_(k=1) ^N k^s )/(1+_− N^r ))=lim_(N→∞) (N^r /(1+_− N^r )).((1+....+N^s )/N^s )=+_− .((r−s)/r)  lim_(x→∞) ((1+_− x^r )/x^r )=+_− 1
$$\mathrm{1}+{N}^{{r}} \sim{N}^{{r}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\underset{{k}=\mathrm{1}} {\overset{{N}} {\sum}}{k}^{{s}} }{{N}^{{s}} }=\underset{{N}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{N}} {\sum}}\frac{\mathrm{1}}{{N}}\left(\frac{{k}}{{N}}\right)^{{s}} =\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{s}} =\frac{\mathrm{1}}{{s}+\mathrm{1}}=\frac{{r}−{s}}{{r}} \\ $$$$\mathrm{1}={r}−{s} \\ $$$${s}+\mathrm{1}={r} \\ $$$$\underset{{N}\rightarrow\infty} {\mathrm{lim}}\frac{\underset{{k}=\mathrm{1}} {\overset{{N}} {\sum}}{k}^{{s}} }{\mathrm{1}\underset{−} {+}{N}^{{r}} }=\underset{{N}\rightarrow\infty} {\mathrm{lim}}\frac{{N}^{{r}} }{\mathrm{1}\underset{−} {+}{N}^{{r}} }.\frac{\mathrm{1}+….+{N}^{{s}} }{{N}^{{s}} }=\underset{−} {+}.\frac{{r}−{s}}{{r}} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}\underset{−} {+}{x}^{{r}} }{{x}^{{r}} }=\underset{−} {+}\mathrm{1} \\ $$

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