A number $N$ is defined as follows: 
\[N=2+22+202+2002+20002+\cdots+2\overbrace{00\ldots000}^{19~0\text{'s}}2\] 
When the value of $N$ is simplified, what is the sum of its digits?

$\textbf{(A) }42\qquad\textbf{(B) }44\qquad\textbf{(C) }46\qquad\textbf{(D) }50\qquad\textbf{(E) }52$
1. **Express each term in the series**:
   Each term in the series can be written in the form \(2 \cdot 10^k + 2\), where \(k\) is the number of zeros between the two 2's. For example:
   - The first term is \(2\).
   - The second term is \(22 = 2 \cdot 10^1 + 2\).
   - The third term is \(202 = 2 \cdot 10^2 + 2\).
   - The fourth term is \(2002 = 2 \cdot 10^3 + 2\).
   - And so on, up to the 20th term, which is \(2 \cdot 10^{19} + 2\).

2. **Sum the series**:
   We need to sum all these terms:
   \[
   N = 2 + (2 \cdot 10^1 + 2) + (2 \cdot 10^2 + 2) + \cdots + (2 \cdot 10^{19} + 2)
   \]
   This can be rewritten as:
   \[
   N = 2 + 2 + 2 + \cdots + 2 + 2 \cdot (10^1 + 10^2 + 10^3 + \cdots + 10^{19})
   \]
   Notice that there are 20 terms of 2, so:
   \[
   N = 20 \cdot 2 + 2 \cdot (10^1 + 10^2 + 10^3 + \cdots + 10^{19})
   \]
   Simplifying the first part:
   \[
   20 \cdot 2 = 40
   \]

3. **Sum the geometric series**:
   The sum of the geometric series \(10^1 + 10^2 + 10^3 + \cdots + 10^{19}\) can be calculated using the formula for the sum of a geometric series:
   \[
   S = a \frac{r^n - 1}{r - 1}
   \]
   where \(a = 10\), \(r = 10\), and \(n = 19\):
   \[
   S = 10 \frac{10^{19} - 1}{10 - 1} = 10 \frac{10^{19} - 1}{9}
   \]

4. **Combine the sums**:
   Now, we combine the sums:
   \[
   N = 40 + 2 \cdot \left(10 \frac{10^{19} - 1}{9}\right)
   \]
   Simplifying inside the parentheses:
   \[
   2 \cdot \left(10 \frac{10^{19} - 1}{9}\right) = \frac{20 (10^{19} - 1)}{9}
   \]
   Therefore:
   \[
   N = 40 + \frac{20 (10^{19} - 1)}{9}
   \]

5. **Sum of the digits**:
   To find the sum of the digits of \(N\), we need to consider the structure of \(N\). The number \(N\) is composed of a series of 2's and 0's. Specifically, it will have 20 digits of 2 and 19 digits of 0. Therefore, the sum of the digits is:
   \[
   20 \cdot 2 = 40
   \]
   Plus the initial 2 from the first term:
   \[
   40 + 2 = 42
   \]

The final answer is \(\boxed{42}\)