Q114 / 160 · AP EAMCET 2024 (21 May Shift 1) · Mathematics

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

  • A
    $\frac{1}{77}$
  • B
    1
  • C
    76
  • Correct answer
    D
    $\frac{1}{78}$

Solution

$\begin{aligned} & \text { } \lim _{n \rightarrow \infty} \frac{1^{77}+2^{77}+\ldots+n^{77}}{n^{78}} \\ & =\lim _{n \rightarrow \infty} \frac{1}{n}\left(\frac{1^{77}}{n^{77}}+\frac{2^{77}}{n^{77}}+\ldots+\frac{n^{77}}{n^{77}}\right) \\ & =\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^n\left(\frac{r}{n}\right)^{77}=\int_0^1 x^{77} d x=\left(\frac{x^{78}}{78}\right)_0^1=\frac{1}{78} \end{aligned}$

Hence, the correct answer is option: 'D'.

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