Nth term test & convergence

In summary, the textbook presents the nth term test for the series \sum (1-3/n)^n with the sum going from 1 to infinity, which shows that it diverges. However, in a previous chapter, the textbook also states a theorem that the limit of (1+x/n)^n converges to e^x for any number x. This can be used to show that the series in the original problem diverges, as (1-3/n)^n converges to e^{-3}, which is not 0.
  • #1
kuahji
394
2
For a problem such as [tex]\sum[/tex] (1-3/n)^n with the sum going from 1 to infinity, the textbook shows to use the nth term test & that it diverges. However in a previous chapter the textbook had a theorem that said the limit of [tex]\sum[/tex] (1+x/n)^n will converge to e^x for any number x.

So I'm confused. Are they using convergence in a different context or what am I missing?
 
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  • #2
It is (1+x/n)^n that converges to e^x, not the sum of that.

And actually, you can use this fact to show that the series in your problem diverges. You know that (1-3/n)^n converges to e^{-3}, which is not 0, hence the series diverges.
 
Last edited:
  • #3
Thanks for the reply. I see that I obviously got confused ^_^.
 

1. What is the Nth term test for convergence?

The Nth term test is a method for determining the convergence or divergence of a series. It states that if the limit of the Nth term of a series is not equal to zero, the series diverges. However, if the limit is equal to zero, further tests are needed to determine convergence or divergence.

2. How is the Nth term test used to determine convergence?

The Nth term test is used by evaluating the limit of the Nth term of a series. If the limit is equal to zero, the series may still converge or diverge. Therefore, further tests, such as the ratio test or the integral test, are needed to determine convergence or divergence.

3. What is the difference between the Nth term test and the ratio test?

The Nth term test evaluates the limit of the Nth term of a series, while the ratio test evaluates the limit of the ratio between consecutive terms. The ratio test is often more effective for determining convergence, as it can also determine absolute and conditional convergence.

4. Can the Nth term test be used for all series?

No, the Nth term test can only be used for series with positive terms. If a series has negative terms, the Nth term test cannot be applied. In such cases, other tests, such as the alternating series test, must be used to determine convergence or divergence.

5. What is the significance of the Nth term test in mathematics?

The Nth term test is an important tool for determining the convergence or divergence of series in mathematics. It is often used in conjunction with other tests to analyze the behavior of series and their limits. It is also a fundamental concept in the study of infinite series and their applications in various branches of mathematics.

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