Assuming $p>0$, we consider the $p$-series $$ 1-\frac{1}{2^p}+\frac{1}{3^p}-\frac{1}{4^p}+\cdots $$ For what $p$-values is the series convergent? For what $p$-values is the series absolutely convergent? Which $p$-values will make it is conditionally convergent?
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$\begingroup$If $p>0$, then the series is convergent by the alternating test (a proof in the link).
If $p>0$, considering that $$ \sum_{n=1}^\infty \left|\frac{(-1)^n}{n^p} \right|=\sum_{n=1}^\infty \frac{1}{n^p} $$ the latter series and the following integral are of the same nature (a proof here) $$ \int_1^\infty\frac{dx}{x^p} $$ and there is a convergence iff $p>1$.
Thus there is a conditional convergence for $0<p\le1$.
$\endgroup$ 0 $\begingroup$we will use the criteria for convergence of alternating series which involves following two conditions 1) (an) must be non-increasing sequence of positive terms 2) limit of an should be 0 as n tends to infinity()
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