# How do you test for convergence given #Sigma (-1)^n(1-1/n^2)# from #n=[1,oo)#?

The series:

is not convergent.

A necessary condition for any series to converge is that the general term of the succession is infinitesimal, that is:

In our case we have:

and so:

is indeterminate.

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To test for convergence of the series (\sum_{n=1}^{\infty} (-1)^n(1-\frac{1}{n^2})), you can use the alternating series test. This test states that if the series alternates in sign and the absolute values of the terms decrease monotonically to zero, then the series converges.

In this series, the terms alternate in sign (((-1)^n)) and the absolute value of each term ((1-\frac{1}{n^2})) decreases monotonically to zero as (n) increases. Therefore, according to the alternating series test, the series converges.

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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.

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