To calculate the sum

\[
A = 1 - \frac{5}{6} + \frac{7}{12} - \frac{9}{20} + \frac{11}{30} - \frac{13}{42} + \frac{15}{56} - \frac{17}{72} + \frac{19}{90},
\]

we can break it down step by step.

1. **Convert each fraction** to have a common denominator if needed, but it might be simpler to calculate each term individually and keep a running total.

2. **Calculate each term** in the series:

- \( 1 = 1 \)
- \( 1 - \frac{5}{6} = \frac{6}{6} - \frac{5}{6} = \frac{1}{6} \)
- \( \frac{1}{6} + \frac{7}{12} = \frac{1}{6} + \frac{14}{24} = \frac{4}{24} + \frac{14}{24} = \frac{18}{24} = \frac{3}{4} \)
- \( \frac{3}{4} - \frac{9}{20} = \frac{15}{20} - \frac{9}{20} = \frac{6}{20} = \frac{3}{10} \)
- \( \frac{3}{10} + \frac{11}{30} = \frac{9}{30} + \frac{11}{30} = \frac{20}{30} = \frac{2}{3} \)
- \( \frac{2}{3} - \frac{13}{42} \) is calculated with a common denominator of \( 126 \):
\[ \frac{2 \cdot 42}{126} - \frac{13 \cdot 3}{126} = \frac{84 - 39}{126} = \frac{45}{126} = \frac{15}{42} = \frac{5}{14} \]
- \( \frac{5}{14} + \frac{15}{56} \) (Common denominator is \( 56 \)):
\[ \frac{5 \cdot 4}{56} + \frac{15}{56} = \frac{20 + 15}{56} = \frac{35}{56} = \frac{5}{8} \]
- \( \frac{5}{8} - \frac{17}{72} \) (Common denominator is \( 72 \)):
\[ \frac{5 \cdot 9}{72} - \frac{17}{72} = \frac{45 - 17}{72} = \frac{28}{72} = \frac{7}{18} \]
- \( \frac{7}{18} + \frac{19}{90} \) (Common denominator is \( 90 \)):
\[ \frac{7 \cdot 5}{90} + \frac{19}{90} = \frac{35 + 19}{90} = \frac{54}{90} = \frac{3}{5} \]

3. **Final result:**

Putting it all together, the final value of \( A \) is

\[
A = \frac{3}{5}.
\]