Sum of series `9/(1!)+19/(2!)+35/(3!)+57/(4!)+...` (A) `7e-3` (B) `12e-5` (C) `16e-5` (D) none

Options

A. `16e-5`
B. `7e-3`
C. `12e-5`
D. `11e-5`

Explanation

To solve the series \( S = \sum_{n=1}^{\infty} \frac{T_n}{n!} \) where \( T_n \) is the \( n \)-th term given by \( T_n = 9 + 10(n-1) + 3(n-1)(n-2) \), we will derive \( T_n \) step by step and then find the sum of the series.

### Step 1: Identify the pattern in the series
The series is given as:
\[ S = \frac{9}{1!} + \frac{19}{2!} + \frac{35}{3!} + \frac{57}{4!} + \ldots \]
We will first find the general term \( T_n \).
...