How do you determine if the series the converges conditionally, absolutely or diverges given #Sigma ((1)^(n+1)n^2)/(n+1)^2# from #[1,oo)#?
The series diverges.
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To determine the convergence properties of the series ( \sum_{n=1}^{\infty} \frac{(1)^{n+1} n^2}{(n+1)^2} ) from ( n = 1 ) to infinity, we'll apply the alternating series test and the ratio test.

Alternating Series Test:
 The series alternates in sign and the absolute values of terms decrease monotonically to zero as ( n ) increases.
 Thus, the series converges conditionally.

Ratio Test:
 Taking the limit as ( n ) approaches infinity of the absolute value of the ratio of consecutive terms: [ \lim_{n \to \infty} \left \frac{a_{n+1}}{a_n} \right = \lim_{n \to \infty} \left \frac{\frac{(1)^{n+2} (n+1)^2}{(n+2)^2}}{\frac{(1)^{n+1} n^2}{(n+1)^2}} \right ] [ = \lim_{n \to \infty} \left \frac{(n+1)^2}{(n+2)^2} \right = 1 ]
 Since the limit is 1, the test is inconclusive.
Therefore, the series ( \sum_{n=1}^{\infty} \frac{(1)^{n+1} n^2}{(n+1)^2} ) 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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