mathop {lim }limits_{n to infty } left[ {frac | vedclass

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(A) ધારો કે $S = \mathop {\lim }\limits_{n 	o \infty } \sum\limits_{r = 1}^n {\frac{n}{{{r^2} + {n^2}}}}$.આપણે સરવાળાને આ રીતે ફરીથી લખી શકીએ:$S = \m

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$\frac{\pi }{4}$

Vedclass · Answer

(A) ધારો કે $S = \mathop {\lim }\limits_{n 	o \infty } \sum\limits_{r = 1}^n {\frac{n}{{{r^2} + {n^2}}}}$.આપણે સરવાળાને આ રીતે ફરીથી લખી શકીએ:$S = \mathop {\lim }\limits_{n 	o \infty } \sum\limits_{r = 1}^n {\frac{n}{{{n^2}(1 + \frac{{{r^2}}}{{{n^2}}})}}} = \mathop {\lim }\limits_{n 	o \infty } \frac{1}{n} \sum\limits_{r = 1}^n {\frac{1}{{1 + {{(\frac{r}{n})}^2}}}}$.નિશ્ચિત સંકલનની વ્યાખ્યાનો ઉપયોગ કરતા,$\mathop {\lim }\limits_{n 	o \infty } \frac{1}{n} \sum\limits_{r = 1}^n {f(\frac{r}{n})} = \int_0^1 {f(x)dx}$,જ્યાં $f(x) = \frac{1}{{1 + {x^2}}}$.તેથી,$S = \int_0^1 {\frac{1}{{1 + {x^2}}}dx}$.સંકલનનું મૂલ્ય શોધતા:$S = [	an^{ - 1} x]_0^1 = 	an^{ - 1}(1) - 	an^{ - 1}(0) = \frac{\pi }{4} - 0 = \frac{\pi }{4}$.

$\mathop {\lim }\limits_{n \to \infty } \left[ {\frac{n}{{1 + {n^2}}} + \frac{n}{{4 + {n^2}}} + \frac{n}{{9 + {n^2}}} + .... + \frac{1}{{2n}}} \right]$ નું મૂલ્ય કેટલું થાય?

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