Rationalise each term and add: 1/(1 + √2) + 1/(√2 + √3) + 1/(√3 + √4) = ?

Rationalise each part separately. Because the conjugate makes each denominator equal to 1, we get 1/(1 + √2) = √2 − 1, 1/(√2 + √3) = √3 − √2 and 1/(√3 + √4) = √4 − √3 = 2 − √3. Adding them, every surd cancels against its neighbour and only the ends survive: √4 − 1 = 2 − 1 = 1. Stopping after the first term leaves √2 − 1 ≈ 0.414, and forgetting that √4 = 2 leaves the answer unsimplified.