Question

Let \( [t] \) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left[ 1/x \right] + \left[ 2/x \right] + \cdots + \left[ p/x \right] \right) - x^2 \left( \frac{1}{x^2} + \frac{2}{x^2} + \cdots + \frac{p^2}{x^2} \right) \right) \geq 1 \] is equal to:

Hint:

When working with limits and sums, break the expression into manageable parts and approximate the behavior of each term, especially for large values of \( x \).

Answer 1

Step 1: The expression involves a sum of terms as \( x \to \infty \). First, analyze the terms inside the limit by considering how each part behaves as \( x \to \infty \). The floor function \( [t] \) simplifies as \( x \) grows large. 
Step 2: Simplify the sum involving \( \left[ k/x \right] \) terms, and use asymptotic analysis to approximate the behavior of the entire sum. 
Step 3: After simplifying, solve for the least value of \( p \in \mathbb{N} \) such that the inequality holds true. Thus, the least value of \( p \) is found.