How do you test the alternating series #Sigma (1)^(n+1)/sqrtn# from n is #[1,oo)# for convergence?
The series:
is then convergent, since:
and
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To test the convergence of the alternating series (\sum_{n=1}^{\infty} \frac{(1)^{n+1}}{\sqrt{n}}), we can use the Alternating Series Test.

Verify if the series satisfies the conditions of the Alternating Series Test:
 The terms of the series alternate in sign.
 The absolute value of each term decreases as (n) increases.
 The limit of the absolute value of the terms approaches zero as (n) approaches infinity.

Check if the series satisfies the conditions:
 The terms alternate in sign since ((1)^{n+1}) alternates between positive and negative.
 The absolute value of each term (\frac{1}{\sqrt{n}}) decreases as (n) increases because (\sqrt{n}) increases as (n) increases.
 The limit of the absolute value of the terms as (n) approaches infinity is zero because (\lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0).
Since the series satisfies all the conditions of the Alternating Series Test, we can conclude that the series converges.
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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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