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Evaluate-lim-n-n-n-2-




Question Number 155295 by mathdanisur last updated on 28/Sep/21
Evaluate:   lim_(n→∞) ((Σ n)/n^2 ) = ?
$$\mathrm{Evaluate}:\:\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\Sigma\:\boldsymbol{\mathrm{n}}}{\boldsymbol{\mathrm{n}}^{\mathrm{2}} }\:=\:? \\ $$
Commented by mathdanisur last updated on 28/Sep/21
Very nice Ser, thank you
$$\mathrm{Very}\:\mathrm{nice}\:\boldsymbol{\mathrm{S}}\mathrm{er},\:\mathrm{thank}\:\mathrm{you} \\ $$
Commented by MJS_new last updated on 28/Sep/21
Σ_(j=1) ^n  j =((n^2 +n)/2)  (((n^2 +n)/2)/n^2 )=(1/2)+(1/(2n))  ⇒ answer is (1/2)
$$\underset{{j}=\mathrm{1}} {\overset{{n}} {\sum}}\:{j}\:=\frac{{n}^{\mathrm{2}} +{n}}{\mathrm{2}} \\ $$$$\frac{\frac{{n}^{\mathrm{2}} +{n}}{\mathrm{2}}}{{n}^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}{n}} \\ $$$$\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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