# How do you determine whether the infinite sequence #a_n=n*cos(n*pi)# converges or diverges?

Let us look at some details.

Let us take the limit of its subsequence of only even terms.

Since the subsequence diverges, the original sequence also diverges.

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To determine whether the infinite sequence ( a_n = n \cos(n\pi) ) converges or diverges, we can analyze its behavior.

- Recall that the cosine function oscillates between -1 and 1.
- For integer values of ( n ), ( \cos(n\pi) ) will be either 1, -1, or 0.
- For ( n ) even, ( \cos(n\pi) ) is 1.
- For ( n ) odd, ( \cos(n\pi) ) is -1.

Now, let's look at the sequence ( a_n = n \cos(n\pi) ):

- When ( n ) is even, ( \cos(n\pi) = 1 ), so ( a_n = n \cdot 1 = n ).
- When ( n ) is odd, ( \cos(n\pi) = -1 ), so ( a_n = n \cdot (-1) = -n ).

This means that the sequence alternates between positive and negative terms, with the absolute values increasing without bound as ( n ) increases.

Since the terms do not approach a fixed value as ( n ) approaches infinity, the sequence ( a_n = n \cos(n\pi) ) diverges.

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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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