Convergence of Infinite Series: A Comparison Test Approach

In summary, the conversation discusses the sums of two infinite series, specifically \sum_{n=1}^{\infty} cos(\frac{\pi}{2n}) and \sum_{n=1}^{\infty} sin(\frac{\pi}{2n}). It is determined that the first series diverges, but there is uncertainty about the convergence or divergence of the second series due to approaching zero terms as n approaches infinity. The suggestion to use a comparison test with the inequality \frac{1}{2}x\leq \sin(x) is given for further investigation.
  • #1
Barbados_Slim
15
0
I was doing my math homework when I started thinking about the sums of two infinite series.
I determined that the sum of the first series [itex]\sum_{n=1}^{\infty} cos(\frac{\pi}{2n}) [/itex] diverges. I could not figure out whether or not the series [itex]\sum_{n=1}^{\infty} sin(\frac{\pi}{2n})[/itex] converges or diverges. I think it diverges but I'm unsure because as n approaches infinity each term in the series approaches zero.
Any help would be much appreciated. Thank you in advance
 
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  • #2
Try to do a comparison test. Try to use the inequality:

[tex]\frac{1}{2}x\leq \sin(x)[/tex]

which holds for small x.
 

FAQ: Convergence of Infinite Series: A Comparison Test Approach

1. What is an infinite series?

An infinite series is a sum of infinitely many terms. Each term can be represented by a function or a sequence of numbers.

2. How do you find the sum of an infinite series?

To find the sum of an infinite series, you can use various methods such as the geometric series test, telescoping series, or the integral test. These methods involve evaluating the limit of the series as the number of terms approaches infinity.

3. Can an infinite series have a finite sum?

Yes, an infinite series can have a finite sum if the terms of the series decrease rapidly enough as the number of terms increases. This is known as a convergent series.

4. What is the difference between a convergent and a divergent series?

A convergent series has a finite sum, while a divergent series has an infinite sum or does not have a sum at all. A divergent series can also be oscillating or alternating between positive and negative values.

5. How is an infinite series used in real-world applications?

Infinite series can be used in various fields such as physics, engineering, and finance to model and approximate real-world phenomena. They can also be used in calculus to calculate integrals and derivatives of functions.

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