# How do you use the limit comparison test to determine whether the following converge or diverge given #sin(1/(n^2))# from n = 1 to infinity?

We can use the small-angle approximation:

This indeterminate result is discouraging, but we can apply L'Hôpital's rule:

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To use the limit comparison test for determining convergence or divergence of the series ( \sum \sin\left(\frac{1}{n^2}\right) ) from ( n = 1 ) to ( \infty ), you need to compare it with a known series that converges or diverges.

Let's denote the given series as ( a_n = \sin\left(\frac{1}{n^2}\right) ).

Choose a series ( b_n ) that you know the convergence or divergence of, and that behaves similarly to ( a_n ) as ( n ) approaches infinity. In this case, a good choice is ( b_n = \frac{1}{n^2} ), as it has a similar form to ( a_n ).

Next, compute the limit:

[ \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\sin\left(\frac{1}{n^2}\right)}{\frac{1}{n^2}} ]

If this limit is a positive finite number, then both series ( \sum a_n ) and ( \sum b_n ) converge or diverge together. If the limit is zero or infinity, the series may converge or diverge independently.

Calculate the limit and determine its value:

[ \lim_{n \to \infty} \frac{\sin\left(\frac{1}{n^2}\right)}{\frac{1}{n^2}} = \lim_{n \to \infty} n^2 \sin\left(\frac{1}{n^2}\right) = 1 ]

Since the limit is a positive finite number (1), we can conclude that ( \sum \sin\left(\frac{1}{n^2}\right) ) and ( \sum \frac{1}{n^2} ) either both converge or both diverge.

Since ( \sum \frac{1}{n^2} ) is a convergent series (it's a special case of the p-series with ( p = 2 )), by the limit comparison test, we can conclude that ( \sum \sin\left(\frac{1}{n^2}\right) ) also 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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