# How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma (-1)^(n+1)/(n+1)^2# from #[1,oo)#?

See below.

Considering that

we have

but

then

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is absolutely convergent.

As:

is convergent, and:

is absolutely convergent.

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To determine the convergence behavior of the series ( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(n+1)^2} ), we can use the alternating series test and the absolute convergence test.

First, let's check for absolute convergence:

[ \sum_{n=1}^{\infty} \left| \frac{(-1)^{n+1}}{(n+1)^2} \right| = \sum_{n=1}^{\infty} \frac{1}{(n+1)^2} ]

This series is a convergent p-series with ( p = 2 ), so it converges absolutely.

Next, let's check for conditional convergence using the alternating series test. The series ( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(n+1)^2} ) alternates signs and the absolute values of its terms decrease monotonically to zero.

Therefore, since the series converges absolutely, it also converges conditionally.

In summary:

- The series converges absolutely because ( \sum_{n=1}^{\infty} \frac{1}{(n+1)^2} ) converges.
- The series also converges conditionally because it satisfies the conditions of the alternating series test.

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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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