How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma (1)^(n+1)/(n+1)# from #[1,oo)#?
First note that:
So our series is not absolutely convergent.
Hence:
is conditionally convergent.
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To determine the convergence properties of the series ( \sum_{n=1}^{\infty} \frac{(1)^{n+1}}{n+1} ), we can use the Alternating Series Test and the Ratio Test.

Alternating Series Test: The series is alternating because it alternates between positive and negative terms. To apply the Alternating Series Test, we need to check two conditions: a. The terms ( \frac{1}{n+1} ) are positive and decreasing. b. The limit of ( \frac{1}{n+1} ) as ( n ) approaches infinity is zero.
Both conditions are satisfied, so the Alternating Series Test tells us that the series converges.

Absolute Convergence: To check for absolute convergence, we examine the series formed by taking the absolute values of the terms: ( \sum_{n=1}^{\infty} \left \frac{(1)^{n+1}}{n+1} \right = \sum_{n=1}^{\infty} \frac{1}{n+1} ).
We know that ( \frac{1}{n+1} ) is a positive, decreasing sequence. However, ( \sum_{n=1}^{\infty} \frac{1}{n+1} ) is a divergent series (harmonic series with one term removed). Therefore, the original series is not absolutely convergent.

Conditional Convergence: Since the original series converges but the absolute value of the series diverges, we can conclude that the series converges conditionally.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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