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Dartboard

Concentric circles with radii 1, \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), ... are drawn. Alternate donut-shaped regions are shaded.
What is the total shaded area?

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If you enjoyed this puzzle, check out Sunday Afternoon Maths XIX,
puzzles about geometry, or a random puzzle.

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