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A first taste of why algebraic decompositions in \(\mathbb{K}(X)\) matter: an identity between rational fractions, evaluated at integers, turns a stubborn-looking sum into a telescoping one.
  1. Show that \(\dfrac{1}{X (X + 1)} = \dfrac{1}{X} - \dfrac{1}{X + 1}\).
  2. Use this identity to compute the partial sum \(\displaystyle\sum_{k = 1}^{n} \dfrac{1}{k (k + 1)}\) in closed form (telescoping).

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