The problem involves two sums, $S_n$ and $T_n$, defined as follows:

$S_n = \sum_{r=1}^{n} \frac{n^2 + nr + r^2}{n^3}$ and $T_n = \sum_{r=0}^{n-1} \frac{n^2 + nr + r^2}{n^3}$

We are asked to determine whether $T_n$ or $S_n$ are greater or less than $\frac{11}{6}$.

Step-by-step Solution:

1. Simplification of $S_n$:

We begin by analyzing $S_n$:

$S_n = \sum_{r=1}^{n} \frac{n^2 + nr + r^2}{n^3}$ Simplifying the terms inside the sum:

$S_n = \sum_{r=1}^{n} \left( \frac{n^2}{n^3} + \frac{nr}{n^3} + \frac{r^2}{n^3} \right)$ This becomes:

$S_n = \sum_{r=1}^{n} \left( \frac{1}{n} + \frac{r}{n^2} + \frac{r^2}{n^3} \right)$ Now, split the sum:

$S_n = \frac{1}{n} \sum_{r=1}^{n} 1 + \frac{1}{n^2} \sum_{r=1}^{n} r + \frac{1}{n^3} \sum_{r=1}^{n} r^2$

Each of these sums can be evaluated:

• $\sum_{r=1}^{n} 1 = n$,
• $\sum_{r=1}^{n} r = \frac{n(n+1)}{2}$,
• $\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}$.

Substitute these values back into the equation for $S_n$:

$S_n = \frac{n}{n} + \frac{1}{n^2} \cdot \frac{n(n+1)}{2} + \frac{1}{n^3} \cdot \frac{n(n+1)(2n+1)}{6}$ Simplify:

$S_n = 1 + \frac{n+1}{2n} + \frac{(n+1)(2n+1)}{6n^2}$

2. Simplification of $T_n$:

Similarly, for $T_n$:

$T_n = \sum_{r=0}^{n-1} \frac{n^2 + nr + r^2}{n^3}$ Following the same steps, we arrive at:

$T_n = 1 + \frac{n-1}{2n} + \frac{(n-1)(2n-1)}{6n^2}$

3. Analysis:

For large values of $n$, both $S_n$ and $T_n$ approach a limit. Comparing these to $\frac{11}{6}$, we find that:

• $S_n$ is slightly greater than $\frac{11}{6}$ for large $n$,
• $T_n$ is slightly less than $\frac{11}{6}$ for large $n$.

Thus, the correct answers are:

• (A) $T_n < \frac{11}{6}$
• (D) $S_n > \frac{11}{6}$

Conclusion:

The correct options are:

• (A) $T_n < \frac{11}{6}$
• (D) $S_n > \frac{11}{6}$

Would you like further clarification on any step or more details on specific aspects of the sums? Here are some related questions to explore:

1. What are other methods to evaluate large summations like $S_n$ and $T_n$?
2. How does the sum $\sum r^2$ grow in comparison to $\sum r$?
3. Can we approximate $S_n$ and $T_n$ for small values of $n$?
4. How would the sums change if we used a different polynomial inside the summation?
5. How do $S_n$ and $T_n$ behave as $n \to \infty$?

Tip: When simplifying summations, always check if common summation formulas can be applied to avoid manual computation of terms.