Analysis of a_n Series: Convergence or Divergence?

In summary, we discussed the conditions for the convergence or divergence of a series, given that a_n is greater than or equal to 0 and the limit of (a_{n+1}/a_n) approaches a constant value c as n approaches infinity. If c is greater than 1, the series diverges, while if c is less than 1, the series converges. We then applied this concept to the series a_n=n!/n^n and used l'Hopital's Rule to find the limit of (n^n)/((n+1)^n), which is equivalent to the limit of (1+1/n)^n. This limit is a famous limit and is equal to 1, so the final limit
  • #1
azatkgz
186
0
Suppose that [tex]a_n\geq 0[/tex] and there is

[tex]\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}=c[/tex]
If c>1,series diverges.
if c<1 series converges.

For [tex]a_n=\frac{n!}{n^n}[/tex]

[tex]\lim_{n\rightarrow\infty}\frac{(n+1)!/(n+1)^{n+1}}{n!/n^n}[/tex]

[tex]\lim_{n\rightarrow\infty}\frac{n^n}{(n+1)^n}[/tex]

Then I used I'Hopital Rule and got answer 1.
 
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  • #2
I think you'd better check your l'Hopital. Your final limit is closely related to the limit of (1+1/n)^n, which is a famous limit and is not one.
 
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  • #3
A,yes I get it.


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

FAQ: Analysis of a_n Series: Convergence or Divergence?

What is an "a_n series" and why is it analyzed for convergence or divergence?

An "a_n series" is a mathematical series in the form of a_n = f(n), where a_n represents the terms of the series and f(n) is a function of n. It is analyzed for convergence or divergence to determine if the series approaches a finite value (convergence) or approaches infinity (divergence).

What is the difference between a convergent and divergent series?

A convergent series has a finite sum, meaning that the terms of the series eventually approach a specific value. In contrast, a divergent series has an infinite sum, meaning that the terms of the series continue to increase without approaching a specific value.

What are some common tests used to determine the convergence or divergence of a series?

Some common tests used to determine the convergence or divergence of a series include the comparison test, the ratio test, and the integral test. These tests involve comparing the given series to a known series or using calculus to evaluate the series.

How do you know if a series is absolutely convergent or conditionally convergent?

A series is absolutely convergent if the absolute values of its terms converge to a finite value. This means that the series will also converge. A series is conditionally convergent if the series itself converges but the absolute values of its terms do not converge. In this case, the series may converge to a finite value or diverge to infinity.

What is the significance of analyzing the convergence or divergence of a series?

Analyzing the convergence or divergence of a series is important in mathematics and science as it allows us to determine the behavior of a series and make predictions based on that behavior. It also helps us understand the underlying patterns and relationships within the series and can be applied to real-world problems and calculations.

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