How do you test the alternating series #Sigma (-1)^n(2^n+1)/(3^n-2)# from n is #[0,oo)# for convergence?

Answer 1

The series is convergent.

A sufficient condition for an alternating series to converge is established by the Leibniz test stating that if:

(i) #lim_(n->oo) a_n = 0#
(ii) #a_(n+1) < a_n# for #n > N#

then the series is convergent.

In our case:

#lim_(n->oo) (2^n+1)/(3^n-2) = lim_(n->oo) ((2/3)^n -1/3^n)/(1-2/3^n) = 0#

so the first condition is satisfied.

Now consider:

#a_(n+1) = ((2^(n+1)+1)/(3^(n+1)-2)) = 2/3 ((2^n+1/2)/(3^n-2/3))#

clearly:

#(2^n+1/2) < (2^n+1)#

and:

#(3^n-2/3) > (3^n-2)#

so that:

#((2^n+1/2)/(3^n-2/3)) <((2^n+1)/(3^n-2))#

and it follows that:

#a_(n+1) = ((2^(n+1)+1)/(3^(n+1)-2)) = 2/3 ((2^n+1/2)/(3^n-2/3)) < 2/3((2^n+1)/(3^n-2)) < a_n#

Thus also the second condition is satisfied and the series os convergent.

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Answer 2

To test the convergence of the alternating series (\sum_{n=0}^{\infty} (-1)^n\frac{2^n+1}{3^n-2}), you can use the Alternating Series Test.

  1. Check the sequence of terms (a_n = \frac{2^n+1}{3^n-2}).
  2. Verify that the terms decrease in absolute value as (n) increases.
  3. Confirm that the limit of the sequence (a_n) as (n) approaches infinity is zero.

If these conditions are met, 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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