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Can a series converge neither pointwise nor uniformly, or are they the only two 'options' for convergence? Clearly uniformly $\implies$ pointwise, but can a series be neither?

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    A series that is not pointwise convergent is simply divergent. I assume that you are talking about series of functions; in that case, let $f_n(x) = x/n$. Then $\sum_{n = 1}^\infty f_n$ is obviously neither uniformly nor pointwise convergent.2012-04-05
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    There is also a concept of convergence in the mean (integral of the square of the difference) - Apostol's Mathematical Analysis sect 9.13 discusses this and gives an example which converges in the mean to $f(0)=0$ on [0,1], but does not converge pointwise anywhere. Essentially the value is 1 on an interval of length $\frac 1{2^k}$ and zero everywhere else. The functions are created by marching the interval along from 0 to 1, and then going back to zero and halving the length.2012-04-05

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