$$\sum_{k=1}^{\infty} \frac{(-1)^{k} k}{\sqrt{(k+1)(k+2)}} $$ I found that absolute series diverges
2026-03-25 06:51:39.1774421499
Explore the series for absolute and conditional convergence
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Since you don't have$$\lim_{k\to\infty}\frac{(-1)^kk}{\sqrt{(k+1)(k+2)}}=0,\tag1$$the series diverges. And you don't have $(1)$ because$$\lim_{k\to\infty}\left\lvert\frac{(-1)^kk}{\sqrt{(k+1)(k+2)}}\right\rvert=1.$$