Question

The sum of the series $1 + 1/2! + 1/4! + \dots \infty$ is:

• A. $e$
• B. $(e+e^{-1})/2$
• C. $(e-e^{-1})/2$
• D. $e^{2}$

Hint:

$(e+e^{-1})/2$ is the definition of the hyperbolic cosine function, $\cosh(1)$.

Answer 1

The Correct Option is B

Step 1: Concept
Use the expansion $e^{x} = 1 + x/1! + x^2/2! + x^3/3! + \dots$.
Step 2: Analysis
$e^{1} = 1 + 1/1! + 1/2! + 1/3! + \dots$
$e^{-1} = 1 - 1/1! + 1/2! - 1/3! + \dots$
Adding the two: $e + e^{-1} = 2(1 + 1/2! + 1/4! + \dots)$.
Step 3: Conclusion
Sum $= (e + e^{-1})/2$.
Final Answer: (B)