To calculate the value of \( P \), we start with the given series:

\[
P = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72}
\]

Let's calculate each term one by one and then sum them:

1. \( \frac{1}{2} = 0.5 \)
2. \( \frac{1}{6} \approx 0.1667 \)
3. \( \frac{1}{12} \approx 0.0833 \)
4. \( \frac{1}{20} = 0.05 \)
5. \( \frac{1}{30} \approx 0.0333 \)
6. \( \frac{1}{42} \approx 0.0238 \)
7. \( \frac{1}{56} \approx 0.0179 \)
8. \( \frac{1}{72} \approx 0.0139 \)

Now, summing these values together:

\[
P \approx 0.5 + 0.1667 + 0.0833 + 0.05 + 0.0333 + 0.0238 + 0.0179 + 0.0139
\]

Calculating this step-by-step:

- \( 0.5 + 0.1667 = 0.6667 \)
- \( 0.6667 + 0.0833 = 0.75 \)
- \( 0.75 + 0.05 = 0.8 \)
- \( 0.8 + 0.0333 = 0.8333 \)
- \( 0.8333 + 0.0238 \approx 0.8571 \)
- \( 0.8571 + 0.0179 \approx 0.8750 \)
- \( 0.8750 + 0.0139 \approx 0.8889 \)

Thus, the total sum is approximately:

\[
P \approx 0.8889
\]

To get a more precise value, we can compute this using a common denominator or a calculator to confirm that:

\[
P \approx \frac{8}{9} \text{ or } 0.888888\ldots
\]

Therefore, we can conclude:

\[
P \approx 0.8889
\]