Step-by-step telescoping summation:
Each term can be decomposed into partial fractions:

$$\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$$

Writing out the sum for $k = 1$ to $9$:

$$S = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \dots + \left(\frac{1}{9} - \frac{1}{10}\right)$$

All intermediate terms cancel out:

$$S = 1 - \frac{1}{10} = \frac{9}{10}$$

• Incorrect: $10/9$ is the reciprocal.
• Correct: Accurately equals $9/10$.
• Incorrect: $5/6$ would be the sum up to 5 terms.
• Incorrect: $6/5$ is invalid.