From what I gather, I can use the alternating series test to check if an alternating series converges or diverges. Let's say, for example I have the series of $(-1)^k\sin (\frac{1}{k})$. I've proved that the series converges with the limit of sin which is equal to $0$, and $f(x)$ of $\sin(\frac{1}{x})$ is a decreasing function.
Now, how can I prove that it is an absolute convergence? Didn't I just prove that as well by taking the alternating part i.e. $(-1)$ and removing it from the series/taking the absolute value of it?
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$\begingroup$If a series of positive terms converges, the corresponding alternating series will converge. The converse is not true.
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