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E20

[0ZB]Let \(a_ n\) be an infinitesimal decreasing positive sequence. Let \(KβŠ† ℝ\) given by

\[ K=\{ 0\} βˆͺ\{ a_ n : nβˆˆβ„•, nβ‰₯ 1\} ~ ; \]

note that it is compact. Study the dimension of \(K\) in these cases:

  • \(a_ n=n^{-πœƒ}\) with \(πœƒ{\gt}0\);

  • \(a_ n=πœƒ^{-n}\) with \(πœƒ{\gt}1\).

Solution 1

[0ZC]

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