$$ Compute the remaining:
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$$ \frac{1}{n(n+2)} = \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+2} \right) $$
Solution: The equation $ |x| + |y| = 4 $ represents a diamond (a square rotated 45 degrees) centered at the origin.
$$ f(3) + g(3) = m + 3m = 4m $$
$$ f(3) + g(3) = m + 3m = 4m $$
$$
$$ a\omega + b = \omega + 3\omega^2 + 1 \quad \ ext{(1)}
This diamond has diagonals of length 8 (horizontal) and 8 (vertical).
Substitute $ a = -2 $ into (1):
$$ Distribute and simplify:
Solution: To find the center, we complete the square for both $ x $ and $ y $ terms.
Plug in $ x = \omega $:
a\omega + b = \omega + 3\omega^2 + 1 \quad \ ext{(1)}
This diamond has diagonals of length 8 (horizontal) and 8 (vertical).
Substitute $ a = -2 $ into (1):
$$ Distribute and simplify:
Solution: To find the center, we complete the square for both $ x $ and $ y $ terms.
Plug in $ x = \omega $:
$$ $$
$$

\boxed{-2x - 2} 9(x^2 - 4x) - 4(y^2 - 4y) = 44 \frac{1}{2} \left( 1 + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \right) $$ \frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right) = \frac{1}{2} \left( \frac{3978 - 103}{2652} \right) = \frac{1}{2} \cdot \frac{3875}{2652} = \frac{3875}{5304} Distribute and simplify:
Solution: To find the center, we complete the square for both $ x $ and $ y $ terms.
Plug in $ x = \omega $:
$$ $$
$$

\boxed{-2x - 2} 9(x^2 - 4x) - 4(y^2 - 4y) = 44 \frac{1}{2} \left( 1 + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \right) $$ \frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right) = \frac{1}{2} \left( \frac{3978 - 103}{2652} \right) = \frac{1}{2} \cdot \frac{3875}{2652} = \frac{3875}{5304} \frac{1}{51} + \frac{1}{52} = \frac{52 + 51}{51 \cdot 52} = \frac{103}{2652} Subtract (1) - (2):
$$ $$ $$ $$
Complete the square:
This is a telescoping series:
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$$
$$

\boxed{-2x - 2} 9(x^2 - 4x) - 4(y^2 - 4y) = 44 \frac{1}{2} \left( 1 + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \right) $$ \frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right) = \frac{1}{2} \left( \frac{3978 - 103}{2652} \right) = \frac{1}{2} \cdot \frac{3875}{2652} = \frac{3875}{5304} \frac{1}{51} + \frac{1}{52} = \frac{52 + 51}{51 \cdot 52} = \frac{103}{2652} Subtract (1) - (2):
$$ $$ $$ $$
Complete the square:
This is a telescoping series:
Add the two expressions:
$$ \boxed{2x^4 - 4x^2 + 3} $$
$$ So $ h(y) = 2y^2 + 1 $.
Then:
$$

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\frac{1}{2} \left( 1 + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \right) $$ \frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right) = \frac{1}{2} \left( \frac{3978 - 103}{2652} \right) = \frac{1}{2} \cdot \frac{3875}{2652} = \frac{3875}{5304} \frac{1}{51} + \frac{1}{52} = \frac{52 + 51}{51 \cdot 52} = \frac{103}{2652} Subtract (1) - (2):
$$ $$ $$ $$
Complete the square:
This is a telescoping series:
Add the two expressions:
$$ \boxed{2x^4 - 4x^2 + 3} $$
$$ So $ h(y) = 2y^2 + 1 $.
Then:
$$

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$$

h(y) = 2(y^2 - 2y + 1) + 4(y - 1) + 3 = 2y^2 - 4y + 2 + 4y - 4 + 3 = 2y^2 + 1 9(x - 2)^2 - 36 - 4(y - 2)^2 + 16 = 44 Solution:
\Rightarrow a = -2 $$
$$
- In the first quadrant: $ x + y = 4 $, from $ (4, 0) $ to $ (0, 4) $.
Now compute the sum:
$$