# How do you determine whether the sequence #a_n=n(-1)^n# converges, if so how do you find the limit?

The sequence diverges.

We can apply the ratio test for sequences:

Suppose that;

So for our sequence;

So our test limit is:

And so the sequence does not converge.

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To determine whether the sequence (a_n = n(-1)^n) converges, you can analyze the behavior of its subsequences. Notice that for even values of (n), (a_n) is positive, and for odd values of (n), (a_n) is negative. This indicates that the sequence oscillates between positive and negative values indefinitely as (n) increases.

Because the sequence oscillates without approaching a specific value, it does not converge.

Therefore, the sequence (a_n = n(-1)^n) does not converge.

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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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