Determining the Radius of Convergence for a Series with Limited Limit Points

In summary, the question is about determining the radius of convergence of the series \sum\limits_{n = 0}^\infty {\left( {3 + \left( { - 1} \right)^n } \right)^n } z^n using the Cauchy-Hadamard criterion. The suggestion is to find the limit superior of the sequence (3+(-1)^n) and use it to calculate the radius of convergence. However, there is confusion about how to apply the criterion when the sequence only has two limit points. It is clarified that the correct sequence should be (2+(-1)^n) which converges to 3. In summary, the radius of convergence is R =
  • #1
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Could someone please help me out with the following? I need to determine the radius of convergence of the following series. It is exactly as given in the question.

[tex]
\sum\limits_{n = 0}^\infty {\left( {3 + \left( { - 1} \right)^n } \right)^n } z^n
[/tex]

The suggestion is to use the Cauchy-Hadamard criterion. The nth coefficient of this series is a_n = (3+(-1)^n)^n which is positive so |a_n|^(1/n) = (3 + (-1)^n). At first thought there are two limit points of the set of points of |a_n|^(1/n), 1 and 3. So the radius of convergence is R = 1/(limpsup(...)) = (1/3) which is the answer that is given.

The problem is that the set of points of the sequence (3 + (-1)^n) only consists of two points, 1 and 3. So how can it have any limit points? (No neighbourhood of either of these two points contains an 'infinite' number of points of the set since there are only two different points.)

Can someone please explain how to do this question properly? Thanks.
 
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  • #2
Are you saying that a 'constant' sequence, like 3 , 3, 3, 3, ..., does not converge? Hmm, given [itex]\epsilon> 0[/itex], how would I find "N" so that if n> N, [itex]|a_x- 3|< \epsilon[/itex]? Looks to me like, since |3- 3|= 0< [itex]\epsilon[/itex], any N would work. 3 is not a "limit point" in the topological sense but it definitely is a limit of that sequence. However, 1 and 3 are NOT subsequential limits of the sequence [itex](3+ (-1)^n)^n[/itex]. When n is odd, we have (3+ (-1)n)= 3- 1= 2 and when n is even (3+ (-1)n)= 3+ 1= 4. The two subsequential limits are 2 and 4.
 
  • #3
Hmm...I mixed up questions from two sources because they looked similar. The coefficient should've had a 2 in place of the 3, my bad.

(2+(-1)^n) alternates between 1 and 3 but I don't see how the Cauchy-Hadamard criterion can be applied here. I suppose that for sequences with a finite number of different terms I only need to work out limits of the subsequences.
 
  • #4
I think you're over complicating this.

[tex]\limsup _{n \to \infty} (2 + (-1)^n) = 3[/tex]
 
  • #5
But the question was about 2+ (-1)n.
 

1. What is the radius of convergence?

The radius of convergence is a mathematical concept used in power series to determine the range of values for which the series will converge. It is denoted by R and is usually a positive real number or infinity.

2. How is the radius of convergence calculated?

The radius of convergence can be calculated using the ratio test, which involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If the limit is less than 1, the series converges, and the value of R can be calculated. If the limit is greater than 1, the series diverges, and R is equal to 0. If the limit is exactly 1, further tests are needed to determine convergence or divergence.

3. What does a larger radius of convergence indicate?

A larger radius of convergence indicates that the power series has a wider range of values for which it will converge. In other words, the series will converge for more values of the variable for which it is defined.

4. Can the radius of convergence be negative?

No, the radius of convergence is always a positive real number or infinity. It represents the distance from the center of the power series to the nearest point where the series diverges.

5. How is the radius of convergence used in real-world applications?

The concept of the radius of convergence is used in various fields of science and engineering, such as physics, chemistry, and economics, to analyze and approximate functions and solve differential equations. It is also used in computer science and data analysis to represent data with power series and make predictions based on the radius of convergence.

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