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Question #40622

Difficulty: 8/10 | 🟢 Free

The Question

Let $x = \sum_{k=1}^\infty \frac{9}{6^k}$ and $y = \sum_{k=1}^\infty \frac{1}{5^k}$ . Define $S_m = \sum_{n=1}^m \sum_{r=1}^n \binom{n}{r} x^r y^n$ . Find the value of $L_1 + L_2$ , where $L_1 = \lim_{m \to \infty} [S_m]$ and $L_2 = \left[ \lim_{m \to \infty} S_m \right]$ , and $[ \cdot ]$ denotes the greatest integer function.

Options

A.
2
B.
3
C.
4
D.
5

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Binomial Coefficient Summation Properties|Infinite GP Convergence|Greatest Integer Function (GIF)

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between the limit of a greatest integer function (GIF) and the GIF of a limit, specifically when a series converges to an integer from below.

Students often incorrectly assume that $\lim [S_m] = [\lim S_m]$ .

By embedding the sequence within a modulated binomial-geometric sum, the problem forces students to first evaluate the exact finite sum $S_m$ .

The 'lethal collision' occurs at the boundary: because the subtractive error term is strictly positive, $S_m$ approaches 2 strictly from below ( $S_m \to 2^-$ ).

Consequently, the sequence of GIFs stabilizes at 1, yielding $L_1 = 1$ , whereas the GIF of the continuous limit is $[2] = 2$ .

Failing to analyze the sign of the remainder term leads to the trap answer of 4.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Binomial Coefficient Summation Properties
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free