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Uniform Convergence of a Series Explained in Plain Words

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Why does it suffice to say that |S-S_n|< epsilon implies that S=sum(n=1 to infinity) a_n is uniformly convergent? I am looking for the reasoning behind it or some reading material on it.

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Solution Summary

We examine and explain the reasoning behing the idea of the uniform convergence of a series.

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A series converges means that the sequence of its partial sums converges.
You can define a sequence s_n = sum_{k=1}^n a_n and if it has a limit, s, then the series converges to this limit, and the sum of the series is s.
The definition of convergence of a sequence is that for every epsilon there is an N such that for all n>N we have |s_n - s| < epsilon.

Now, if the general term of the sequence or a ...

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