<s>[INST] Absolute convergence of $\sum_{n=1}^{\infty} (-1)^n (1-\cos1/n)$ I would like to see if $$\sum_{n=1}^{\infty} (-1)^n \left(1-\cos \frac{1}{n}\right)$$ converges absolutely. 
 [/INST] Donote your sum by $S$. Taylor series for $\cos$ gives us $\cos x = 1 - \frac{1}{2}x^2 + o(x^3)$, hence $1 - \cos x = \frac{1}{2}x^2 + o(x^3)$. Since $|(-1)^n (1-\cos\frac{1}{n})| = \cos\frac{1}{n}$, it follows that
$$S \leq \sum_{n=1}^{\infty}\left(\frac{1}{n^2} + |o(\frac{1}{n^3})|\right).$$
Since $\frac{o(x^3)}{x^3}\to 0 $, when $x\to 0$, we know that $o(x^3)\leq x^3$ for all $|x|\leq x_0$ for some $x_0\in (0,\infty)$. It follows that
$$S\leq \sum_{n = 1}^{\infty} \frac{1}{n^2} + \sum_{n = 1}^{n_0}o(\frac{1}{n^3}) + \sum_{n = n_0 + 1}^{\infty}\frac{1}{n^3}<\infty$$
where $n_0$ is a integer for which $\frac{1}{n_0}\leq x_0$.
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