For a ∈ R, |a| > 1, let _ n → ∞ ( 1 + 23 + + n3 n ^ 7 / 3 ( 1 (an +...

Let $S = \lim_{n \to \infty} \left(\dfrac{1 + {2}^{\dfrac{1}{3}} + {3}^{\dfrac{1}{3}} + \ldots + {n}^{\dfrac{1}{3}}}{{n}^{\dfrac{7}{3}}\left(\dfrac{1}{{\left(an + 1\right)}^{2}} + \ldots + \dfrac{1}{{\left(an + n\right)}^{2}}\right)}\right)$
$⇒ S = \lim_{n \to \infty} \dfrac{\mathop{∑}_{r = 1}^{n}{r}^{\dfrac{1}{3}}}{{n}^{\dfrac{7}{3}}\mathop{∑}_{r = 1}^{n}\dfrac{1}{{\left(\text{ na } + r\right)}^{2}}}$$<br/>$S = \lim_{n \to \infty} \dfrac{\left(\mathop{∑}_{r = 1}^{n}{\left(\dfrac{r}{n}\right)}^{\dfrac{1}{3}}\dfrac{1}{n}\right) \times {n}^{\dfrac{1}{3}}}{{n}^{\dfrac{7}{3}}\mathop{∑}_{r = 1}^{n}\dfrac{1}{{\left(a + \dfrac{r}{n}\right)}^{2}}\dfrac{1}{{n}^{2}}}$<br/>$S = \lim_{n \to \infty} \dfrac{\mathop{∑}_{r = 1}^{n}{\left(\dfrac{r}{n}\right)}^{\dfrac{1}{3}} . \dfrac{1}{n}}{\mathop{∑}_{r = 1}^{n}{\left(\dfrac{1}{a + \dfrac{r}{n}}\right)}^{2}\dfrac{1}{n}}$<br/>$S = \dfrac{\mathop{∫}_{0}^{1}{x}^{\dfrac{1}{3}}dx}{\mathop{∫}_{0}^{1}\dfrac{1}{{\left(a + x\right)}^{2}} . dx}$<br/>$S = \dfrac{{\left[\dfrac{{x}^{\dfrac{4}{3}}}{\dfrac{4}{3}}\right]}_{0}^{1}}{{\left[- \dfrac{1}{x + a}\right]}_{0}^{1}}$<br/>$S = \dfrac{\dfrac{3}{4}}{\dfrac{1}{a} - \dfrac{1}{a + 1}} = 54$<br/>$\dfrac{1}{a\left(a + 1\right)} = \dfrac{1}{72}$<br/>${a}^{2} + a - 72 = 0$<br/>$a = 8$or$a = - 9