How do you test the alternating series #Sigma (-1)^(n+1)/sqrtn# from n is #[1,oo)# for convergence?

Answer 1
A sufficient condition for an alternating series #sum (-1)^na_n# to converge is that:
(1) #lim a_n= 0#
(2) #a_(n+1) <= a_n#

The series:

#sum_(n=1)^oo (-1)^(n+1)/sqrt(n)#

is then convergent, since:

#lim 1/sqrt(n) = 0#

and

#1/sqrt(n+1) < 1/sqrt(n)#
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Answer 2

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.

  1. 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.
  2. 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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Answer from HIX Tutor

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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