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.
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Choose a simpler series whose convergence behavior is well-known. In this case, consider the series (\sum \frac{1}{n^2}).
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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}]
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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.
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Since the limit is finite and positive, and not equal to zero, both series either converge or diverge together.
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Since the series (\sum \frac{1}{n^2}) is known to converge (by the p-series test with (p = 2)), by the limit comparison test, the given series 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.
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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- How do you apply the ratio test to determine if #Sigma (3^n(n!)^2)/((2n)!)# from #n=[1,oo)# is convergent to divergent?
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