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

Answer 1

The sequence diverges.

We can apply the ratio test for sequences:

Suppose that;

# L=lim_(n rarr oo) |a_(n+1)/a_n| < 1 => lim_(n rarr oo) a_n = 0#
i.e. if the absolute value of the ratio of successive terms in a sequence #{a_n}# approaches a limit #L#, and if #L < 1#, then the sequence itself converges to #0#. It is important to note that this is a statement about the convergence of the sequence #{a_n}#, and it is not a statement about the series #sum a_n#.

So for our sequence;

# a_n = n(-1)^n #

So our test limit is:

# L = lim_(n rarr oo) | ( (n+1)(-1)^(n+1) ) / ( n(-1)^n ) | # # \ \ \ = lim_(n rarr oo) | ( (n+1)(-1)^n(-1) ) / ( n(-1)^n ) | # # \ \ \ = lim_(n rarr oo) | ( (n+1)(-1) ) / ( n ) | # # \ \ \ = lim_(n rarr oo) | ( (n+1) ) / ( n ) | # # \ \ \ = lim_(n rarr oo) | 1+1/n | # # \ \ \ > 1 #

And so the sequence does not converge.

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Answer 2

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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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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