Series Convergence: Exploring the Limit and Root Test Methods

  • Thread starter azatkgz
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In summary, the series \sum_{n=1}^{\infty}\left(e-\left(1+\frac{1}{n}\right)^n\right)^p converges for all p<1. The solution was found using the Root Test and rewriting the series in terms of the natural logarithm and using the limit as n approaches infinity. The series was then simplified to show that it converges for all p<1.
  • #1
azatkgz
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0

Homework Statement


Determine whether the series converges or diverges.


[tex]\sum_{n=1}^{\infty}\left(e-\left(1+\frac{1}{n}\right)^n\right)^p[/tex] where p is a parameter[/tex]


The Attempt at a Solution



[tex]\lim_{n\rightarrow\infty}e-\left(1+\frac{1}{n}\right)^n=0[/tex]

so by using Root Test i decided that

[tex]\limsup_{n\rightarrow\infty}\sqrt[n]{\left(e-\left(1+\frac{1}{n}\right)^n\right)^p}<1[/tex]
Which gives that series converges
 
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  • #2
What if p=0, for example?
 
  • #3
Oh,I see.Limit is 1.I must try another way.
 
  • #4
[tex]\sum_{n=1}^{\infty}\left(e-\left(1+\frac{1}{n}\right)^n\right)^p=\sum_{n=1}^{\infty}\left(e-e^{n\ln \left(1+\frac{1}{n}\right)}\right)^p=\sum_{n=1}^{\infty}\left(e-e^{n\left(\frac{1}{n}-\frac{1}{2n^2}+O(\frac{1}{n^3})\right)}\right)^p=\sum_{n=1}^{\infty}\left(e-e^{1-\frac{1}{2n}+O(\frac{1}{n^2})}\right)^p[/tex]

[tex]=\sum_{n=1}^{\infty}e^p\left(1-e^{-\frac{1}{2n}+O(\frac{1}{n^2})}\right)^p=\sum_{n=1}^{\infty}e^p\left(1-(1-\frac{1}{2n}+O(\frac{1}{n^2}))\right)^p[/tex]

[tex]=\sum_{n=1}^{\infty}e^p\left(\frac{1}{2n}+O(\frac{1}{n^2})\right)^p[/tex]

Is it converges for all p<1?
 

1. What is the limit test method for series convergence?

The limit test method is a mathematical test used to determine the convergence or divergence of a series. It involves taking the limit of the terms in the series and seeing if it approaches a finite value or infinity. If the limit is a finite value, the series is said to converge. If the limit is infinity, the series is said to diverge.

2. How is the root test method used to determine series convergence?

The root test method is another mathematical test used to determine the convergence or divergence of a series. It involves taking the nth root of the absolute value of each term in the series and seeing if the resulting series converges. If the resulting series converges, then the original series also converges. If the resulting series diverges, then the original series may or may not converge. Further tests may be needed to determine convergence or divergence.

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

Absolute convergence refers to when a series converges regardless of the order in which the terms are added. Conditional convergence, on the other hand, refers to when a series only converges when the terms are added in a specific order. In other words, the terms of a conditionally convergent series can be rearranged in a different order to make the series diverge.

4. Can the limit and root test methods be used for all series?

No, the limit test method and root test method can only be used for certain types of series. For example, they can be used for series with positive terms, alternating series, and some power series. Other types of series may require different methods for determining convergence or divergence.

5. Are there any other methods for determining series convergence besides the limit and root tests?

Yes, there are several other tests and methods for determining series convergence. Some examples include the ratio test, the integral test, and the comparison test. Each test has its own criteria and may be more suitable for certain types of series.

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