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Peano’s Unnoticed Proof of Borel’s Theorem

Ádám Besenyei
The American Mathematical Monthly
Vol. 121, No. 1 (January 2014), pp. 69-72
DOI: 10.4169/amer.math.monthly.121.01.069
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Page Count: 4
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Peano’s Unnoticed Proof of Borel’s Theorem
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Abstract In 1876, P. du Bois-Reymond gave an example of an infinitely differentiable function whose Taylor series diverges everywhere except one point. By generalizing this example, G. Peano proved a theorem in 1884, often credited to É. Borel, which states that every power series is a Taylor series of some infinitely differentiable function. The aim of the paper is to recall Peano’s unnoticed contributions to this result.

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