Does sin(4/3n) Converge or Diverge?

  • Thread starter mattmannmf
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In summary, the series E sin(4/3n) can be determined to diverge by comparing it to the series 3/(4n) and noting that the latter is greater than the former. This shows that sin(4/3n) must also diverge. Algebraic methods such as the direct comparison test or limit comparison test may not be applicable in this case due to the presence of the sine function.
  • #1
mattmannmf
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given the series, determine whether the series converges or diverges:
(E is my sigma)

E sin(4/3n)


Now would it just be the same if i compared 1/n to 3/4n to diverge

Then could I just say since 3/4n diverges then sin(3/4n) must also diverge?

I wouldn't really know how to do the algebra for the direct comparison test or limit comparison test with the sin being there. But i can do the algebra for comparing 1/n with 3/4n..help please
 
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1. What does it mean for a series to converge or diverge?

Convergence and divergence refer to the behavior of an infinite series as more terms are added. A series that converges has a finite limit as the number of terms approaches infinity, while a series that diverges has no finite limit.

2. How can I determine if a series converges or diverges?

There are several tests that can be used to determine the convergence or divergence of a series, including the comparison test, the ratio test, and the integral test. These tests involve evaluating the behavior of the terms in the series as well as their relationships to known convergent or divergent series.

3. What is the significance of a series converging or diverging?

The convergence or divergence of a series has important implications in mathematics, physics, and engineering. In mathematics, it is essential for understanding infinite sequences and the concept of infinity. In physics and engineering, it is used to analyze and model real-world phenomena that involve infinite sums, such as electrical circuits, fluid dynamics, and signal processing.

4. Can a series both converge and diverge?

No, a series can only either converge or diverge. It is not possible for a series to both approach a finite limit and have no finite limit as the number of terms increases.

5. Is there a way to determine the exact value of a convergent series?

Yes, if a series is known to converge, there are techniques such as the geometric series formula and the telescoping series method that can be used to find the exact value of the series. However, not all convergent series have a known closed-form solution, in which case numerical methods may be used to approximate the value.

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