Solution: Simplify $ p_n = \frac{n}{n(n^2 + 2)} = \frac{1}{n^2 + 2} $. As $ n \to \infty $, $ n^2 + 2 \to \infty $, so $ \lim_{n \to \infty} p_n = \boxed{0} $.
["### Simplify $ p_n = \frac{n}{n(n^2 + 2)} $: A Clear and Powerful Mathematical Insight", "In mathematical analysis, simplifying expressions not only makes them easier to work with but also reveals deeper structural insights—especially when analyzing limits. One such expression that often appears in calculus, number theory, and algebra is:", "$$\np_n = \frac{n}{n(n^2 + 2)}\n$$", "At first glance, this fraction may seem complex, but with a simple algebraic simplification, we uncover a clean and elegant form. Let’s explore how to simplify this expression and understand its behavior as $ n \ o \infty $.", "---", "#### Step 1: Simplify the Expression", "Start by examining the numerator and denominator:", "$$\np_n = \frac{n}{n(n^2 + 2)}\n$$", "Notice that $ n $ appears in both the numerator and the denominator. Since $ n <br/>\ne 0 $, we can safely cancel $ n $ from numerator and denominator:", "$$\np_n = \frac{1}{n^2 + 2}\n$$", "This simplified form is far easier to analyze, particularly when studying limits or asymptotic behavior.", "---", "#### Step 2: Analyze the Limit as $ n \ o \infty $", "Now that we have $ p_n = \frac{1}{n^2 + 2} $, consider what happens as $ n \ o \infty $. Since $ n^2 \ o \infty $, the denominator grows without bound:", "$$\n\lim_{n \ o \infty} p_n = \lim_{n \ o \infty} \frac{1}{n^2 + 2} = \frac{1}{\infty} = 0\n$$", "Thus,\n$$\n\boxed{\lim_{n \ o \infty} p_n = 0}\n$$", "This limiting behavior confirms that $ p_n $ approaches zero asymptotically. Even though $ p_n $ remains positive and well-defined for all positive integers $ n $, its magnitude diminishes indefinitely as $ n $ increases.", "---", "#### Why This Simplification Matters", "Simplifying $ p_n $ from its original form reveals underlying patterns useful in both theoretical and applied contexts:", "- Ease of Evaluation: The simplified $ \frac{1}{n^2 + 2} $ allows straightforward computation of values and limits.\n- Asymptotic Insight: It clearly demonstrates why the sequence converges to zero, illustrating a common pattern in sequences where linear numerators dominate quadratic growth.\n- Generalizability: Such simplifications help extend techniques to more complex expressions by reducing them to familiar building blocks.", "---", "#### Conclusion", "Simplifying mathematical expressions is not just a matter of aesthetics—it’s a fundamental tool in uncovering truth and facilitating deeper understanding. The transformation $ \frac{n}{n(n^2 + 2)} = \frac{1}{n^2 + 2} $ exemplifies how a moment of careful analysis transforms complexity into clarity. And knowing that $ \lim_{n \ o \infty} p_n = 0 $ equips us with essential knowledge for sequences, limits, and convergence.", "---", "Key Takeaway:\n$$\np_n = \frac{n}{n(n^2 + 2)} = \frac{1}{n^2 + 2} \Rightarrow \lim_{n \ o \infty} p_n = \boxed{0}\n$$", "Simplify to understand, simplify to analyze—this is mathematics in action."]
