How do you test the convergence of the series #cos(n) sin (pi/n)^2#?
is absolutely convergent by direct comparison.
Use the inequalities:
so that:
As:
is absolutely convergent by direct comparison.
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To test the convergence of the series (\cos(n) \sin\left(\frac{\pi}{n}\right)^2), you can use the limit comparison test or the ratio test. Let's use the limit comparison test.

Choose a simpler series whose convergence behavior is wellknown. In this case, consider the series (\sum \frac{1}{n^2}).

Take the limit of the ratio of the terms of the given series to the terms of the chosen simpler series as (n) approaches infinity.
[\lim_{n \to \infty} \frac{\cos(n) \sin\left(\frac{\pi}{n}\right)^2}{\frac{1}{n^2}}]
 Simplify and evaluate this limit.
[= \lim_{n \to \infty} \frac{n^2 \cos(n) \sin\left(\frac{\pi}{n}\right)^2}{1}]

Since (\cos(n)) is bounded between (1) and (1), and (\sin\left(\frac{\pi}{n}\right)^2) approaches (0) as (n) approaches infinity, the limit simplifies to 0.

Since the limit is finite and positive, and not equal to zero, both series either converge or diverge together.

Since the series (\sum \frac{1}{n^2}) is known to converge (by the pseries test with (p = 2)), by the limit comparison test, the given series also converges.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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