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

The series converges

Perform the ratio test

Therefore,

Then,

As,

We conclude that the series converges ( absolutely) by the ratio test

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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.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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