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## Necessary condition for the uniform convergence of a certain series

Let be a decreasing sequence of positive numbers such that the series

is uniformly convergent. Then must satisfy .

Note that this result implies that the series is not uniformly convergent on . It is surprisngly similar to the following result:

Suppose that is a decreasing sequence of positive real numbers such that the series is convergence. Then . It is no surprise that the proofs of these two results are similar.

*Proof:* If the series converges uniformly, then the following series

converges uniformly to zero. Picking we find that for . Therefore

and summing up we get

The fact that converges uniformly to zero implies that .

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Categories: Analysis, Fourier Analysis, Fourier Series
Analysis, trick, uniform convergence

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