Menu Close

lim-n-1-1-n-3-4-8-n-3-9-27-n-3-n-2-2n-3-




Question Number 162336 by mnjuly1970 last updated on 28/Dec/21
  lim_( n→∞)  ((1/(1+n^( 3) )) +(( 4)/(8 +n^( 3) )) + (9/(27 +n^( 3) )) +...+(n^( 2) /(2n^( 3) )) )=?
$$ \\ $$$${lim}_{\:{n}\rightarrow\infty} \:\left(\frac{\mathrm{1}}{\mathrm{1}+{n}^{\:\mathrm{3}} }\:+\frac{\:\mathrm{4}}{\mathrm{8}\:+{n}^{\:\mathrm{3}} }\:+\:\frac{\mathrm{9}}{\mathrm{27}\:+{n}^{\:\mathrm{3}} }\:+…+\frac{{n}^{\:\mathrm{2}} }{\mathrm{2}{n}^{\:\mathrm{3}} }\:\right)=? \\ $$$$ \\ $$
Answered by mindispower last updated on 28/Dec/21
lim_(n→∞) Σ_(k=1) ^n (k^2 /(n^3 +k^3 ))=lim_(n→∞) (1/n)Σ_(k=1) ^n ((((k/n))^2 )/(1+((k/n))^3 ))  =∫_0 ^1 ((x^2 dx)/(1+x^3 ))=((ln(2))/3)
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{k}^{\mathrm{2}} }{{n}^{\mathrm{3}} +{k}^{\mathrm{3}} }=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(\frac{{k}}{{n}}\right)^{\mathrm{2}} }{\mathrm{1}+\left(\frac{{k}}{{n}}\right)^{\mathrm{3}} } \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{2}} {dx}}{\mathrm{1}+{x}^{\mathrm{3}} }=\frac{{ln}\left(\mathrm{2}\right)}{\mathrm{3}} \\ $$
Commented by mnjuly1970 last updated on 29/Dec/21
   grateful sir power
$$\:\:\:{grateful}\:{sir}\:{power} \\ $$
Commented by mindispower last updated on 29/Dec/21
pleasur sir have a great Day
$${pleasur}\:{sir}\:{have}\:{a}\:{great}\:{Day} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *