How do you determine the convergence or divergence of #Sigma (-1)^(n+1)cschn# from #[1,oo)#?

Question

Ezekiel Alexander · Accepted Answer

The series converges

Luke Alvarez · Answer

undefined To determine the convergence or divergence of the series $ \sum_{n=1}^{\infty} (-1)^{n+1} \csc(n) $ from $ n = 1 $ to $ n = \infty $, we analyze the behavior of its terms.

The series $ \sum_{n=1}^{\infty} (-1)^{n+1} \csc(n) $ is an alternating series, where each term alternates in sign. The term $ \csc(n) $ represents the cosecant function evaluated at $ n $. As $ n $ approaches infinity, $ \csc(n) $ approaches $ 0 $.

The alternating series test states that if a series is alternating and the absolute value of its terms decreases monotonically to $ 0 $, then the series converges. In this case, the terms $ \csc(n) $ approach $ 0 $ as $ n $ approaches infinity.

However, the convergence of the series $ \sum_{n=1}^{\infty} (-1)^{n+1} \csc(n) $ is challenging to determine conclusively due to the erratic behavior of the cosecant function. It's known that $ \csc(n) $ oscillates between $ -\infty $ and $ \infty $ for different values of $ n $, and its behavior is not strictly decreasing toward $ 0 $.

Therefore, the convergence or divergence of the series $ \sum_{n=1}^{\infty} (-1)^{n+1} \csc(n) $ cannot be definitively determined using conventional convergence tests. Additional methods, such as more specialized convergence tests or numerical approximation, may be required to determine its convergence behavior.