Find the partial fraction decomposition for 1/x(x+1) and use the result to find the following sum:

Verified step by step guidance

1. Step 1: Start by finding the partial fraction decomposition of \( \frac{1}{x(x+1)} \). Assume \( \frac{1}{x(x+1)} = \frac{A}{x} + \frac{B}{x+1} \).
2. Step 2: Multiply both sides by \( x(x+1) \) to clear the denominators: \( 1 = A(x+1) + Bx \).
3. Step 3: Expand and combine like terms: \( 1 = Ax + A + Bx \).
4. Step 4: Group the terms involving \( x \): \( 1 = (A + B)x + A \).
5. Step 5: Set up the system of equations by comparing coefficients: \( A + B = 0 \) and \( A = 1 \). Solve for \( A \) and \( B \).