Help with testing the convergence of a series

In summary, the conversation discusses the convergence and divergence of the series 1+2r+r2+2r3+r4+2r5+... for different values of r, and the use of the nth root test to determine convergence. The possibility of removing the first term and still showing convergence is also mentioned.
  • #1
bolzano
15
0
Hi i have to show that the series 1+2r+r2+2r3+r4+2r5+... converges for r=[itex]\frac{2}{3}[/itex] and diverges for r=[itex]\frac{4}{3}[/itex] using the nth root test.

The sequence [itex]\sqrt[n]{a_{n}}[/itex]comes a bit complicated so i was wondering if I can remove the 1st term a1=1 and show that 2r+r2+2r3+r4+2r5+... converges, using the test of course.

Am i doing something wrong?

Thanks :)
 
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  • #2
Yes you can remove any finite number of terms and not change whether the series converges or not.
 
  • #3
So [itex](\sqrt[n]{a_{n}})[/itex][itex]=2r,r,\sqrt[3]{2}r,r[/itex],[itex]\sqrt[5]{2}r,...[/itex] which converges to [itex]r[/itex] right? :)
 

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

Convergence in a series refers to the behavior of the terms in a sequence as they approach a particular value. In other words, it is the process of determining whether a series of numbers will eventually settle on a finite value or continue to increase or decrease indefinitely.

2. How do you test the convergence of a series?

There are several methods for testing the convergence of a series, including the ratio test, root test, integral test, and comparison test. These methods involve analyzing the behavior of the terms in the series as they approach a limit or comparing the series to a simpler known series.

3. What is the purpose of testing the convergence of a series?

Testing the convergence of a series is important in mathematics and science because it allows us to determine the behavior of a sequence of numbers and make predictions about its future values. It also helps us to determine whether a series is convergent or divergent, which has implications for its use in calculations and equations.

4. What happens if a series does not converge?

If a series does not converge, it is considered to be divergent. This means that the terms in the sequence do not approach a finite value, but instead continue to increase or decrease without bound. In some cases, this may indicate an error in calculations or assumptions, while in others it may have a specific mathematical meaning.

5. Are there any real-world applications for testing the convergence of a series?

Yes, there are many real-world applications for testing the convergence of a series. In finance, for example, series convergence is used in the calculation of compound interest and investment returns. In physics and engineering, series convergence is used in the analysis of electrical circuits and the behavior of waves. In statistics, it is used in the analysis of data and the prediction of future trends.

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