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# Test for convergence.$\sum_{j=1}^{\infty}(-1)^j \sin \left(\frac{\pi}{4 j}\right)$

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In this task, we need to check the convergence of the given series. Here, the ratio test is inconclusive. Thus, we can use the alternating series test to check the convergence. The alternating series test states that $\sum_{n=1}^{\infty}(-1)^na_n$ converges if

$1)$ the $a_n$'s are all positive.

$2)$ $a_n \ge a_{n+1}$ for all $n\ge N.$

$3)$ $\lim_{n \to \infty}a_n=0.$

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