Q.3 The sum of the following infinite series is:
1/1! +1/2! +1/3! +1/4! +1/5! + …
The infinite series ∑n=1∞ 1/n! sums to e−1, where e≈2.71828 is the base of the natural logarithm. This result comes from the Taylor series expansion of ex evaluated at x=1, excluding the n=0 term.
✅ Correct Answer: (C) e − 1
The full exponential series is e = ∑n=0∞ 1/n! = 1 + 1/1! + 1/2! + 1/3! + ⋯. Subtracting the n=0 term (which is 1) gives ∑n=1∞ 1/n! = e−1.
Option Explanations
Series Derivation
The Taylor series for ex is ex = ∑n=0∞ xn/n!. At x=1, e = 1 + ∑n=1∞ 1/n!. The series converges rapidly due to factorial growth in the denominator.
Partial sum up to n=5:
1 + 0.5 + 0.1667 + 0.04167 + 0.008333 ≈ 1.7167
nearing e – 1 ≈ 1.71828
1 + 0.5 + 0.1667 + 0.04167 + 0.008333 ≈ 1.7167
nearing e – 1 ≈ 1.71828
