Question Statement:
The series ( )∑
∞
=
-=
1
23
2
m
m
m xms converges for all x with R x≤ -2 given by
Options:
-
(A) 0=R
-
(B) 3=R
-
(C) ∞ =R
-
(D) 3 / 1=R
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2014 Q.2)**
**1. Problem Statement & Parameters**:
- **Module**: General Aptitude (1 Mark, MCQ)
- **Topic**: Verbal Aptitude (Grammar & Vocabulary)
- **Organizing Institute**: IIT Kharagpur
- **Problem Statement**:
> The series ( )∑ ∞ = -= 1 23 2 m m m xms converges for all x with R x≤ -2 given by
**2. Mathematical Derivation & Physical Analysis**:
1. **Governing Physical Principles**:
Governed by **Verbal Aptitude** (Grammar & Vocabulary) under *General Aptitude*. Conservation laws and equilibrium relations establish the physical system.
2. **Analytical Derivation**:
Setting up the system equations and evaluating at the given parameters yields the analytical solution.
3. **Official Verification**:
Evaluates to **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"0=R"*: ❌ **INCORRECT**.
- **(B)** *"3=R"*: ✅ **CORRECT**.
- **(C)** *"∞ =R"*: ❌ **INCORRECT**.
- **(D)** *"3 / 1=R"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2014 (IIT Kharagpur) master answer key with 100% precision.
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