Limits, Continuity & Differentiability
Limits of Series
Grade 12
Question:
<p>The value of <span>\(\lim_{n \to \infty} \frac{1 \cdot n + 2 \cdot (n-1) + 3 \cdot (n-2) + \ldots + n \cdot 1}{1^2 + 2^2 + \ldots + n^2}\)</span> is</p>
<p>(a) 1</p>
<p>(b) -1</p>
<p>(c) \(\frac{1}{2}\)</p>
<p>(d) \(\frac{1}{2}\)</p>
Step-by-Step Solution
Key Concept: Evaluate the numerator as $\sum_{k=1}^{n} k(n+1-k)$ and the denominator using the formula $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$; simplify.
<p>No solution provided in the page text.</p>
Correct Answer: C