For an interesting example, take $\sum_{n=1}^\infty \frac{|\sin(n)|^n}{n}$. Deciding whether or not this converges seems to require more knowledge than is currently available about the rational approximations of $\pi$. The series $\sum_{n=1}^\infty \frac{|\sin(n t \pi)|^n|}{n}$ pi)|^n}{n}$ converges for almost every real $t$ (in the sense of Lebesgue measure), but diverges for $t$ in a dense $G_\delta$ subset of $\mathbb R$.
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For an interesting example, take $\sum_{n=1}^\infty \frac{|\sin(n)|^n}{n}$. Deciding whether or not this converges seems to require more knowledge than is currently available about the rational approximations of $\pi$. The series $\sum_{n=1}^\infty \frac{|\sin(n t \pi)|^n|}{n}$ converges for almost every real $t$ (in the sense of Lebesgue measure), but diverges for $t$ in a dense $G_\delta$ subset of $\mathbb R$. |
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