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Question #41880

Difficulty: 8/10 | 🟢 Free

The Question

Let $\mathbb{I}(P)$ be the indicator function which equals $1$ if the proposition $P$ is true, and $0$ otherwise. Evaluate the following limit: $L = \lim_{n \to \infty} \frac{1}{n^4} \sum_{k=1}^n k^3 \cdot \mathbb{I}(\gcd(k, 210) = 1)$

Options

A.
$\frac{2}{35}$
B.
$\frac{8}{35}$
C.
$\frac{1}{4}$
D.
\text{The limit does not exist}

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Sigma Notation Properties|Indicator Function Logic|The L'Hôpital's Rule Misuse

Why This Question Is Tricky

Common Mistakes

This question tests the misuse of the Stolz-Cesàro theorem (the discrete analogue of L'Hôpital's Rule).

Students often attempt to evaluate $\lim \frac{a_n}{b_n}$ by finding $\lim \frac{a_n - a_{n-1}}{b_n - b_{n-1}}$ .

Here, the ratio of differences is asymptotically $\frac{1}{4}\mathbb{I}(\gcd(n, 210)=1)$ , which oscillates between $0$ and $1/4$ .

A common trap is to falsely conclude that since the limit of differences does not exist, the original limit does not exist.

However, the converse of Stolz-Cesàro is invalid; the sequence of averages converges perfectly using inclusion-exclusion or Möbius inversion.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Sigma Notation Properties
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free