GATE 2025 PH – Question 3
The sum of the following infinite series is: $$\frac1{1!}+\frac1{2!}+\frac1{3!}+\frac1{4!}+\frac1{5!}+\cdots$$
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Correct answer: (C) $e-1$
Explanation
The exponential series is
$$e=\sum_{n=0}^{\infty}\frac1{n!}=\frac1{0!}+\frac1{1!}+\frac1{2!}+\frac1{3!}+\cdots=1+\frac1{1!}+\frac1{2!}+\cdots$$
The series in the question starts at $n=1$, so it is the whole series for $e$ minus the first term $\dfrac1{0!}=1$:
$$\frac1{1!}+\frac1{2!}+\frac1{3!}+\cdots=e-1\quad(\text{option C}).$$
(Numerically $1+0.5+0.1667+0.0417+0.0083+\cdots\approx1.718=e-1$.)