Determine if series converges or diverges

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In summary, the two given series Ʃ cos^2(n)/(n^2+8) and Ʃ 5n/(n^2+1) * cos(2πn) have different convergence properties. The first series diverges due to the oscillating nature of the cosine function, while the second series converges due to the boundedness of the cosine function. The convergence or divergence of infinite series should not be determined based on intuition, but rather through the use of theorems and series tests. In this case, the first series can be shown to diverge using the comparison test, while the second series can be shown to converge using the limit comparison test. It is important to have a solid understanding of
  • #1
turbokaz
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Homework Statement


Ʃ cos^2(n)/(n^2+8)

Ʃ 5n/(n^2+1) * cos(2πn)


Homework Equations





The Attempt at a Solution


I think that both series diverge. Can anyone validate this or tell me if I'm wrong?
 
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  • #2
Why do you think they diverge?
 
  • #3
I'm changing my mind. First one diverges because the cosine will oscillate between -1 and 1. The second one converges because it will always go to 0?
 
  • #4
Convergence and divergence of infinite series is a counterintuitive and complicated matter. Don't base you're conclusion of guesses and intuition. Use theorems instead. Do you know any? Which ones may be useful for those series?
 
  • #5
turbokaz said:
I'm changing my mind. First one diverges because the cosine will oscillate between -1 and 1. The second one converges because it will always go to 0?

You have cos2n which will always be positive since it's squared. As a result, cos2n ≤ 1 and so cos2n/(n2 + 8) ≤ 1/(n2 + 8). From there it's not too difficult to show whether it converges or diverges using one of the series tests.

Before looking at the second one, which series tests are you familiar with?
 

1. What does it mean for a series to converge or diverge?

Convergence and divergence refer to the behavior of a sequence of numbers as more terms are added. A convergent series approaches a specific finite value as the number of terms increases, while a divergent series does not have a finite limit and may either approach infinity or oscillate between values.

2. How do you 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, ratio test, and integral test. These tests involve comparing the given series to a known convergent or divergent series or using mathematical calculations to evaluate the behavior of the series.

3. What is the difference between absolute and conditional convergence?

Absolute convergence occurs when a series converges regardless of the order in which the terms are added. Conditional convergence occurs when a series only converges if the terms are added in a specific order. This distinction is important when using certain tests, such as the rearrangement test.

4. Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent. If a series is neither convergent nor divergent, it is said to be oscillatory or divergent with oscillation.

5. How is the convergence or divergence of a series related to its sum?

If a series is convergent, its sum can be calculated by finding the limit of the series as the number of terms approaches infinity. However, if a series is divergent, its sum is undefined and cannot be calculated using traditional methods.

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