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

Answer 1

the series is indeterminate.

We can easily see that the series is not convergent, since:

#lim_(n->oo) (-1)^n != 0#

We can take a closer look at the partial sums:

#sum_(n=1)^oo (-1)^n#
#s_1 = -1# #s_2 =0# #s_3 = -1# #...#

and we can prove by induction that:

#{(s_(2n) = 0),(s_(2n+1) = -1):}#

so that partial sums oscillate between the two values and do not converge to a limit.

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Answer 2

To test the alternating series (\sum_{n=1}^{\infty} (-1)^n) for convergence, you can use the Alternating Series Test. This test states that if the terms of the series alternate in sign, decrease in absolute value, and approach zero as (n) approaches infinity, then the series converges.

So, to test the series:

  1. Check if the terms alternate in sign.
  2. Check if the absolute values of the terms decrease as (n) increases.
  3. Check if the limit of the absolute values of the terms as (n) approaches infinity is zero.

If all three conditions are met, then the alternating 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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