lim _{n rightarrow infty} frac{3}{n} left{ 4 | vedclass

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(D) આપેલ પદાવલિ $\lim _{n \rightarrow \infty} \frac{3}{n} \sum _{r=0}^{n-1} \left( 2 + \frac{r}{n} \right)^2$ છે. નિયત સંકલનની વ્યાખ્યા મુજબ,$\lim _{n

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$19$

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(D) આપેલ પદાવલિ $\lim _{n \rightarrow \infty} \frac{3}{n} \sum _{r=0}^{n-1} \left( 2 + \frac{r}{n} \right)^2$ છે. નિયત સંકલનની વ્યાખ્યા મુજબ,$\lim _{n \rightarrow \infty} \frac{1}{n} \sum _{r=0}^{n-1} f\left( \frac{r}{n} \right) = \int _0^1 f(x) dx$. અહીં,આ પદાવલિને $3 \int _0^1 (2+x)^2 dx$ તરીકે લખી શકાય. ધારો કે $u = 2+x$,તો $du = dx$. જ્યારે $x=0, u=2$ અને જ્યારે $x=1, u=3$. તેથી,$3 \int _2^3 u^2 du = 3 \left[ \frac{u^3}{3} \right] _2^3 = [u^3] _2^3 = 3^3 - 2^3 = 27 - 8 = 19$.

$\lim _{n \rightarrow \infty} \frac{3}{n} \left\{ 4 + \left( 2 + \frac{1}{n} \right)^2 + \left( 2 + \frac{2}{n} \right)^2 + \dots + \left( 3 - \frac{1}{n} \right)^2 \right\}$ ની કિંમત શોધો.

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