lim _{n rightarrow infty}left[frac{1^k+2^k+3^ | vedclass

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(C) हमारे पास है,$\lim _{n \rightarrow \infty}\left[\frac{1^k+2^k+3^k+\ldots+n^k}{n^{k+1}}\right]$ $= \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r

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$\frac{1}{k+1}$

Vedclass · Answer

(C) हमारे पास है,$\lim _{n \rightarrow \infty}\left[\frac{1^k+2^k+3^k+\ldots+n^k}{n^{k+1}}\right]$ $= \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} \left(\frac{r}{n}\right)^k$ यह योगफल की सीमा के रूप में निश्चित समाकलन की परिभाषा है: $\int_0^1 x^k \, dx$ समाकलन का मान ज्ञात करने पर: $\int_0^1 x^k \, dx = \left[ \frac{x^{k+1}}{k+1} \right]_0^1$ $= \frac{1^{k+1}}{k+1} - \frac{0^{k+1}}{k+1} = \frac{1}{k+1}$ अतः,सीमा का मान $\frac{1}{k+1}$ है.

$\lim _{n \rightarrow \infty}\left[\frac{1^k+2^k+3^k+\ldots+n^k}{n^{k+1}}\right]=$

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