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Given the series: $$\sum_{k=1}^{\infty}\frac{\ln(k)}{k\cdot 3^k}x^k$$

I found that the interval of convergence is $(-3,3)$ where the endpoints diverge. So is the function continuous at the endpoint?

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a part of definition of continuity, the function in question has to be well-defined at the point of interest. As you are saying, at the endpoints, the function diverges, hence is not well-defined, and thus cannot be continuous.

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