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I want to construct a complex power series with the radius of convergence $R=1$ that diverges in:

1) $k$ points on the circle $\{|z|=1\}$.

2) countable number of points on the circle

I have already crafted series that converges on the whole circle and diverges on the whole circle.

Can anybody help with the problem?

Update: I have googled a series $\sum{\frac{z^{kn}}{kn}}$ that is an answer for 1). Though I still need help for the countable case.

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    what were your functions that converge and diverge on the whole circle?2012-03-21

2 Answers 2

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For part 1), we can use Abel's test to prove $\log(1-z)$ converges everywhere on the boundary except at $z=1$. I believe now taking $f(z) = \log(1-z^k)$ should do.

For part 2)

The generating function for the number of partitions of $n$ has a countable number of singularities and to deal with that, Hardy, Ramanujan and Littlewood came up with the Circle Method, which they used to determine the asymptotic behaviour of the partition number.

But, I suppose there is an easier example, as this is homework.

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Hint: for (2), take a sequence of points $z_k$ on the circle converging to $1$, say, and try putting together series that diverge at each $z_k$.