Evaluation of Limit of Sums Involving Radicals and Reciprocal Squares

To find the possible values of $a \in \mathbb{R}$ with $|a| > 1$, we consider the given limit:

$\lim_{n \to \infty} \left( \frac{\sum_{r=1}^n r^{1/3}}{n^{7/3} \sum_{r=1}^n \frac{1}{(an+r)^2}} \right) = 54$

We can rewrite the expression in terms of Riemann sums as $n \to \infty$.

1. Evaluating the Numerator

The sum in the numerator can be written as: $\sum_{r=1}^n r^{1/3} = n^{1/3} \sum_{r=1}^n \left(\frac{r}{n}\right)^{1/3} = n^{4/3} \left( \frac{1}{n} \sum_{r=1}^n \left(\frac{r}{n}\right)^{1/3} \right)$

As $n \to \infty$, the Riemann sum converts to the definite integral: $\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^n \left(\frac{r}{n}\right)^{1/3} = \int_{0}^{1} x^{1/3} \, dx = \left[ \frac{3}{4} x^{4/3} \right]_0^1 = \frac{3}{4}$

Thus, the numerator is asymptotically: $\sum_{r=1}^n r^{1/3} \sim \frac{3}{4} n^{4/3}$

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2. Evaluating the Denominator

The denominator is: $n^{7/3} \sum_{r=1}^n \frac{1}{(an+r)^2} = n^{7/3} \sum_{r=1}^n \frac{1}{n^2 \left(a + \frac{r}{n}\right)^2} = n^{1/3} \sum_{r=1}^n \frac{1}{\left(a + \frac{r}{n}\right)^2} = n^{4/3} \left( \frac{1}{n} \sum_{r=1}^n \frac{1}{\left(a + \frac{r}{n}\right)^2} \right)$

Since $|a| > 1$, the function $\frac{1}{(a+x)^2}$ is continuous on $[0, 1]$. As $n \to \infty$, this Riemann sum converts to: $\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^n \frac{1}{\left(a + \frac{r}{n}\right)^2} = \int_{0}^{1} \frac{1}{(a+x)^2} \, dx = \left[ -\frac{1}{a+x} \right]_0^1 = -\frac{1}{a+1} - \left(-\frac{1}{a}\right) = \frac{1}{a(a+1)}$

Thus, the denominator is asymptotically: $n^{7/3} \sum_{r=1}^n \frac{1}{(an+r)^2} \sim \frac{1}{a(a+1)} n^{4/3}$

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3. Calculating the Limit

Substituting the asymptotic expressions back into the original limit: $\lim_{n \to \infty} \frac{\frac{3}{4} n^{4/3}}{\frac{1}{a(a+1)} n^{4/3}} = \frac{3}{4} a(a+1)$

We are given that this limit equals $54$: $\frac{3}{4} a(a+1) = 54$ $a(a+1) = 54 \times \frac{4}{3} = 72$ $a^2 + a - 72 = 0$ $(a + 9)(a - 8) = 0$

This yields: $a = -9 \quad \text{or} \quad a = 8$

Both solutions satisfy the condition $|a| > 1$.

Therefore, the possible values of $a$ are $-9$ and $8$, which correspond to options (A) and (D).