Question Statement:
The sum of the following infinite series is:
1
1! + 1
2! + 1
3! + 1
4! + 1
5! + ⋯
Options:
-
(A) 𝜋
-
(B) 1 + e
-
(C) e - 1
-
(D) e
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.3)**
**1. Mathematical Formulation & Taylor Series**:
- Given infinite series:
$S = \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \dots = \sum_{n=1}^{\infty} \frac{1}{n!}$
**2. Step-by-Step Derivation**:
- The Taylor series expansion of the exponential function $e^x$ evaluated at $x = 1$ is:
$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$
- Substituting $x = 1$:
$e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \dots = 1 + S$
- Subtracting $1$ from both sides:
$S = e - 1$
**3. Official Verification**:
The verified official IIT answer is **Option (C)** (`e - 1`).
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