For a \in R (the set of all real numbers), a \neq – 1 \underset{n \rightarrow \infty}{Lim} \textrm{ } \frac{\left(\right. 1^{a} + 2^{a} + \ldots \ldots . + n^{a} \left.\right)}{\left(\left(\right. n + 1 \left.\right)\right)^{\textrm{ } a - 1} \textrm{ } \left[\right. \left(\right. na + 1 \left.\right) + \left(\right. na + 2 \left.\right) + \ldots \ldots . + \left(\right. na + n \left.\right) \left]\right.} = \frac{1}{60} . Then a =

Engineering

Mathematics

limit as a sum

Question

For a $\in$ R (the set of all real numbers), a $\neq$ – 1

$\underset{n \to \infty}{Lim} \frac{(1^{a} + 2^{a} + \dots \dots . + n^{a})}{{(n + 1)}^{a - 1} [(na + 1) + (na + 2) + \dots \dots . + (na + n)]} = \frac{1}{60}$ . Then a =

$\frac{- 15}{2}$

$\frac{- 17}{2}$

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$\underset{n \to \infty}{Lim} \frac{n^{a} [{(\frac{1}{n})}^{a} + {(\frac{2}{n})}^{a} + \dots \dots . + {(\frac{n}{n})}^{a}]}{n^{a} {(1 + \frac{1}{n})}^{a - 1} [(n + \frac{1}{a}) + (n + \frac{2}{a}) + \dots \dots . (a + \frac{n}{n})]} = \frac{1}{60}$

$\Rightarrow \underset{n \to \infty}{Lim} \frac{\sum_{r = 1}^{n} {(\frac{r}{n})}^{a} \cdot \frac{1}{n}}{\sum_{r = 1}^{n} (n + \frac{r}{a}) \cdot \frac{1}{n}} = \frac{1}{60} \Rightarrow \frac{\int_{0}^{1} x^{a} dx}{\int_{0}^{1} (n + x) dx} = \frac{1}{16} \Rightarrow \frac{\frac{1}{(a + 1)} {[x^{a + 1}]}_{0}^{1}}{{[nx + \frac{x^{2}}{2}]}_{0}^{1}} = \frac{1}{16}$

$∴ a = 7 and b = \frac{- 17}{2}$