Sequence Limits and the Squeeze Theorem

Problem Statement

Let the sequence $u_n$ be defined by:

$u_n = \frac{1^2}{n^3 + 1} + \frac{2^2}{n^3 + 2} + \dots + \frac{n^2}{n^3 + n}$

for integers $n \ge 1$. You may assume the standard result: $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$. (Do NOT prove this).

(i) Explain why, for all integers $k$ such that $1 \le k \le n$:

$\frac{k^2}{n^3 + n} \le \frac{k^2}{n^3 + k} \le \frac{k^2}{n^3 + 1}$

(ii) Hence, evaluate $\lim_{n \to \infty} u_n$.

Hints

• Part (i): Consider how varying the denominator alters the overall value of a fraction. Since $k$ is positive, compare the sizes of $n^3+1$, $n^3+k$, and $n^3+n$.
• Part (ii): Apply the summation symbol $\sum_{k=1}^n$ across the inequality established in part (i). Factor out the denominators, substitute the given sum of squares formula, and apply the Squeeze Theorem to the outer bounds.

Solutions

• Part (i): For $1 \le k \le n$, the denominators satisfy $n^3 + 1 \le n^3 + k \le n^3 + n$. Taking the reciprocal reverses the inequality (as all terms are positive): $\frac{1}{n^3 + n} \le \frac{1}{n^3 + k} \le \frac{1}{n^3 + 1}$. Multiplying through by the positive term $k^2$ yields the required result: $\frac{k^2}{n^3 + n} \le \frac{k^2}{n^3 + k} \le \frac{k^2}{n^3 + 1}$.
• Part (ii): Summing the inequalities from $k=1$ to $n$ gives: $\sum_{k=1}^n \frac{k^2}{n^3 + n} \le u_n \le \sum_{k=1}^n \frac{k^2}{n^3 + 1}$. Factoring out the denominators (since they are independent of $k$) yields $\frac{1}{n^3 + n} \sum_{k=1}^n k^2 \le u_n \le \frac{1}{n^3 + 1} \sum_{k=1}^n k^2$. Substituting the given sum formula gives $\frac{n(n+1)(2n+1)}{6(n^3+n)} \le u_n \le \frac{n(n+1)(2n+1)}{6(n^3+1)}$. As $n \to \infty$, the highest power of $n$ in both bounding expressions is $n^3$, leaving the ratio of the leading coefficients: $\frac{2}{6} = \frac{1}{3}$. Since both bounds approach $\frac{1}{3}$, by the Squeeze Theorem, $\lim_{n \to \infty} u_n = \frac{1}{3}$.

Takeaways

• Bounding Strategies: Establishing inequalities by manipulating denominators is a fundamental technique for rigorous sequence problems, particularly well-suited for Extension 2 level assessments.
• Summation Mechanics: Recognizing that denominators independent of the summation index $k$ can be factored out of the sigma notation is a critical step that prevents algebraic clutter and confusion.
• Limit Theorems: This style of question seamlessly bridges the algebraic manipulation of series with formal limit evaluation using the Squeeze Theorem.