Determine whether the series is convergent or divergent.

In summary, convergence in a series means the sum of its terms approaches a finite value as the number of terms increases, while divergence means the sum grows without bound. To determine convergence or divergence, various tests such as the ratio test or root test can be used. A series cannot be both convergent and divergent. Real-life applications of determining convergence or divergence include finance and engineering. There are special cases where convergence or divergence cannot be determined, requiring specialized tests.
  • #1
Sabricd
27
0
Hello,

I have to determine whether the series converges or diverges.

It is [tex]\Sigma[/tex] (-1)^n * cos(Pi/n) where n=1 and goes to infinity.

First I took the absolute value of the function and got the limit from n to infinity of cos(pi/n) and as a result I got 1 because cos(0)=1. However my textbook says it's divergent. Could you please help me understand why it is divergent?

Thank you!
 
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  • #2
Nevermind! I figured out why. I guess I confused the alternating series test with some other one. :)

Thanks!
 

1. What is the definition of convergence and divergence in a series?

The convergence of a series means that the sum of all the terms in the series approaches a finite value as the number of terms increases. In contrast, divergence of a series means that the sum of the terms in the series grows without bound as the number of terms increases.

2. How do you determine if a series is convergent or divergent?

To determine if a series is convergent or divergent, you can use various tests such as the ratio test, root test, or comparison test. These tests involve evaluating the limit of the series and comparing it to known values or using other mathematical techniques.

3. Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent. It cannot be both at the same time. A series is considered convergent if its sum approaches a finite value, and it is considered divergent if its sum grows without bound.

4. What are some real-life applications of determining convergence or divergence in a series?

Determining convergence or divergence in a series has many real-life applications, such as in finance, economics, and engineering. For example, in finance, determining the convergence or divergence of a series can help forecast future stock prices or analyze investment portfolios. In engineering, it can help predict the behavior of a system or structure under different conditions.

5. Are there any special cases where the convergence or divergence of a series cannot be determined?

Yes, there are some special cases where the convergence or divergence of a series cannot be determined, such as when the terms of the series are alternating or when the terms are not strictly decreasing. In these cases, specialized tests may be needed to determine the convergence or divergence of the series.

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