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If we were given a sequence of positive numbers {a_n}, can we construct a smooth function(can be derivatived for infinitely many times) that has nth derivatives at 0 that is bigger than the given number a_n? I believe it's right, but I cannot construct one to convince others.

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up vote 1 down vote accepted

Borel's theorem gives more: you can choose the value at $0$. Moreover, we can choose the values of the derivatives on a set which consitst of isolated points.

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