How do you determine if the series the converges conditionally, absolutely or diverges given #sum_(n=1)^oo (-1)^(n+1)arctan(n)#?
The series diverges.
The Alternating Series Test
And if the series doesn't converge conditionally then it doesn't converge absolutely.
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To determine if the series ( \sum_{n=1}^{\infty} (-1)^{n+1}\arctan(n) ) converges conditionally, absolutely, or diverges, you would typically follow these steps:
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Check if the series is alternating. In this case, the series ( \sum_{n=1}^{\infty} (-1)^{n+1}\arctan(n) ) is alternating because it alternates in sign.
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Check if the series satisfies the conditions for the Alternating Series Test, which states that if the terms of an alternating series decrease in absolute value and approach zero as ( n ) increases, then the series converges.
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Determine if the series ( \sum_{n=1}^{\infty} |\arctan(n)| ) converges or diverges. This will help determine if the original series converges absolutely or not.
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If the series ( \sum_{n=1}^{\infty} |\arctan(n)| ) converges, then the original series ( \sum_{n=1}^{\infty} (-1)^{n+1}\arctan(n) ) converges absolutely.
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If the series ( \sum_{n=1}^{\infty} |\arctan(n)| ) diverges, but ( \lim_{n \to \infty} \arctan(n) = 0 ), then apply the Alternating Series Test to the original series to determine if it converges conditionally.
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If the series ( \sum_{n=1}^{\infty} (-1)^{n+1}\arctan(n) ) satisfies the conditions of the Alternating Series Test, then it converges conditionally.
In summary:
- Check if the series is alternating.
- Check if the terms decrease in absolute value and approach zero.
- Determine if the series ( \sum_{n=1}^{\infty} |\arctan(n)| ) converges.
- Apply the Alternating Series Test if necessary to determine conditional convergence.
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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.
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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