# How do you determine the convergence or divergence of #Sigma ((-1)^(n+1))/(2n-1)# from #[1,oo)#?

Leibniz's rule, or the alternating series test, can be used to determine if one of these series converges or not.

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Arranging and adding successive terms

so

and we have

so

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To determine the convergence or divergence of the series ( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2n-1} ) from ( n = 1 ) to ( \infty ), you can use the Alternating Series Test. This test states that if a series has alternating terms and meets two conditions: 1) the terms decrease in magnitude as ( n ) increases, and 2) the limit of the terms approaches zero as ( n ) approaches infinity, then the series converges. In this case, the terms ( \frac{(-1)^{n+1}}{2n-1} ) alternate in sign and decrease in magnitude as ( n ) increases. Additionally, the limit of the terms as ( n ) approaches infinity is zero. Therefore, by the Alternating Series Test, the series converges.

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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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