lim _{n} {rightarrow infty}left[left(1+frac{1 | vedclass

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(B) ધારો કે $V = \lim _{n \rightarrow \infty}\left[\prod_{k=1}^{n} \left(1+\frac{k^2}{n^2}\right)\right]^{1 / n}$.બંને બાજુ પ્રાકૃતિક લઘુગણક લેતા:$\lo

Vedclass · Accepted Answer

$e^{\frac{\pi}{2}-2}$

Vedclass · Answer

(B) ધારો કે $V = \lim _{n ightarrow \infty}\left[\prod_{k=1}^{n} \left(1+\frac{k^2}{n^2}ight)ight]^{1 / n}$.બંને બાજુ પ્રાકૃતિક લઘુગણક લેતા:$\log V = \lim _{n ightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \log \left(1+\frac{k^2}{n^2}ight)$.આ $\int_{0}^{1} f(x) dx$ સ્વરૂપનો રીમાન સરવાળો છે જ્યાં $f(x) = \log(1+x^2)$:$\log V = \int_{0}^{1} \log(1+x^2) dx$.ખંડશઃ સંકલનનો ઉપયોગ કરતા,$\int \log(1+x^2) dx = x \log(1+x^2) - 2x + 2 	an^{-1}(x)$.$0$ થી $1$ સુધી મૂલ્ય શોધતા:$\log V = \log 2 - 2 + \frac{\pi}{2}$.તેથી,$V = e^{\log 2 - 2 + \frac{\pi}{2}} = 2 e^{\frac{\pi}{2}-2}$.

$\lim _{n}$ ${\rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right) \ldots \left(1+\frac{n^2}{n^2}\right)\right]^{1 / n}=$

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