lim _{n rightarrow infty} frac{1^{77}+2^{77}+ | vedclass

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(D) We are given the limit: $\lim _{n \rightarrow \infty} \frac{1^{77}+2^{77}+\ldots+n^{77}}{n^{78}}$This can be rewritten as: $\lim _{n \rightarrow \

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$\frac{1}{78}$

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(D) We are given the limit: $\lim _{n \rightarrow \infty} \frac{1^{77}+2^{77}+\ldots+n^{77}}{n^{78}}$This can be rewritten as: $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} \left(\frac{r}{n}\right)^{77}$Using the definition of a definite integral as the limit of a Riemann sum,$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} f\left(\frac{r}{n}\right) = \int_{0}^{1} f(x) dx$Here,$f(x) = x^{77}$.Therefore,the limit becomes: $\int_{0}^{1} x^{77} dx$Evaluating the integral: $\left[ \frac{x^{78}}{78} \right]_{0}^{1} = \frac{1^{78}}{78} - \frac{0^{78}}{78} = \frac{1}{78}$

$\lim _{n \rightarrow \infty} \frac{1^{77}+2^{77}+\ldots+n^{77}}{n^{78}} = $

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