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I need to find one (or more) interesting complex power series to give to my students for their analysis exam. Ideally, this would be a power series that has interesting behavior at the boundary, i.e. does not converge everywhere/nowhere, but only at select points. To check this, they have at their disposal Abel's criterion, Dirichlet's criterion and Weierstrass' M-test. The classic examples (that they've seen) are of course the ones with coefficients $1, \frac{1}{n}$, and $\frac{1}{n^2}$. Others seem hard to find.

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How about considering series such as

$$z^4 + \frac12 z^8 + \frac13 z^{12} + \dots ?$$

This would converge everywhere on the unit circle apart from the points $\pm 1$ and $\pm i$, which can easily be seen from the fact that it is obtained from the series $z + \frac12 z^2 + \frac13 z^3 + \dots$, by replacing $z$ with $z^4$, so that the single "bad point" at $z=1$ becomes the four "bad points" where $z^4=1$.

It should be easy to invent other examples along these lines where the series does not converge at a finite set of points on the circle, but the general case of looking at exactly which subsets of the circle of convergence can be realised as sets of points where a power series diverges seems to be a tough problem, as seen here on MO.

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