I have the following series defined.
$$\displaystyle\sum_{k=1}^{n} \cos \left( {\frac{\pi}{2}} k \right) \frac{k}{k+1000} \frac{1}{\sqrt{k}}$$ where $n = 1,2...$
How to test whether this series converges or diverges ?
2026-03-31 21:50:12.1774993812
Series convergence or divergence how to test
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We have: $$\sum_{n=1}^{+\infty}\cos\left(\frac{\pi k}{2}\right)\frac{k}{k+1000}\frac{1}{\sqrt{k}} = \sum_{n=0}^{+\infty}(-1)^n \frac{2n+1}{2n+1001}\frac{1}{\sqrt{2n+1}}$$ hence the series is convergent by Leibniz' criterion, since $\frac{2n+1}{2n+1001}\frac{1}{\sqrt{2n+1}}$ decreases towards zero.