Question

\[ 1 + \frac{3}{1!} + \frac{5}{2!} + \frac{7}{3!} + \cdots \infty \]

• A. 5
• B. \(2e\)
• C. \(3e\)
• D. None of these

Hint:

Use standard expansions of \(e^x\) for factorial series.

Answer 1

The Correct Option is C

Concept: \[ \sum \frac{(2n+1)}{n!} = 2\sum \frac{n}{n!} + \sum \frac{1}{n!} \]

Step 1: Split series.
\[ = \sum \frac{1}{n!} + 2\sum \frac{n}{n!} \]

Step 2: Use identities.
\[ \sum \frac{1}{n!} = e,\quad \sum \frac{n}{n!} = e \]

Step 3: Final.
\[ = e + 2e = 3e \]