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Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers

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Infinite sums of reciprocals of certain figurate numbers are often discussed in a first-year calculus course. For example, the sum of the reciprocals of the triangular numbers is easily determined since this is an example of a telescoping series. Moreover, the exact value of the sum of the reciprocals of the squares is often mentioned in such courses (with the exact value being 2/6). Unfortunately, little appears in the literature about sums of reciprocals of figurate numbers with more than four sides. Our goal in this capsule is to obtain a closed formula for all such infinite series where the number of sides of the figurate numbers is even and at least six.

Document Type: Miscellaneous

Publication date: November 1, 2008

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