How do you test the convergence of the series #cos(n) sin (pi/n)^2#?

Answer 1

#sum_(n=0)^oo cos(n)sin^2(pi/n)#

is absolutely convergent by direct comparison.

Use the inequalities:

#abs(cosx) <= 1#
#abs(sin x) <= abs (x)#

so that:

#abs(cos(n)sin^2(pi/n)) <= pi^2/n^2#

As:

#sum_(n=0)^oo pi^2/n^2 = pi^2 sum_(n=0)^oo 1/n^2# is convergent based on the p-series test, then also:
#sum_(n=0)^oo cos(n)sin^2(pi/n)#

is absolutely convergent by direct comparison.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

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.

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

  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}}]

  1. Simplify and evaluate this limit.

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

  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.

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

  3. 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.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7