Does the Complex Series Sum of (n!)^3/(3n)! * z^n Diverge?

Therefore, the series converges for |z| < 27 and the radius of convergence is R = 27.In summary, the given series \displaystyle\sum_{n =
  • #1
fauboca
158
0
[tex]\displaystyle\sum_{n = 1}^{\infty}\frac{(n!)^3}{(3n)!}z^n , \ z\in\mathbb{C}[/tex]

By the ratio test,
[tex]
\displaystyle L = \lim_{n\to\infty}\left|\frac{[(n + 1)!]^3 z^{n + 1} (3n)!}{[3(n + 1)]! (n!)^3 z^n}\right| = \lim_{n\to\infty}\left|\frac{z (n + 1)^2}{3}\right| = \infty.[/tex]

Therefore, the series diverges and there is no radius of convergence.

Correct?
 
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  • #2
i don't think your cancellation is quite correct 3(n+1)=3n+3
 
  • #3
lanedance said:
i don't think your cancellation is quite correct 3(n+1)=3n+3

Thanks, I see the problem.

[tex]|z|<27[/tex]

So [itex]R = 27[/itex] then
 
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  • #4
can you show how you got there?
 
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  • #5
lanedance said:
can you show how you got there?

[tex]\lim_{n\to\infty}\left|\frac{z (n + 1)^3}{(3n + 3)(3n + 2)(3n + 1)}\right|[/tex]


[tex]\Rightarrow \lim_{n\to\infty}|z|\left(\frac{n^3}{27n^3}\right) = \frac{1}{27}|z| < 1 \Rightarrow |z| < 27[/tex]
 
  • #6
yeah looks good

you can just just take the result below if you like , rather than carrying the z through though its good to understand why
[tex] r = \lim_{n \to \infty} \left|\frac{c_n}{c_{n+1}}\right| [/tex]
 

FAQ: Does the Complex Series Sum of (n!)^3/(3n)! * z^n Diverge?

What is the definition of divergence of a complex series?

The divergence of a complex series refers to the behavior of the series as the number of terms in the series approaches infinity. If the series does not approach a finite limit as the number of terms increases, it is said to diverge.

How is the divergence of a complex series determined?

The divergence of a complex series can be determined by applying various tests, such as the comparison test, ratio test, or root test. These tests examine the behavior of the terms in the series and can determine if the series converges or diverges.

What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series that converges regardless of the order of the terms. In contrast, conditional convergence refers to a series that only converges when the terms are arranged in a specific order. A series can be absolutely convergent but not conditionally convergent.

Can a complex series converge and diverge simultaneously?

No, a complex series cannot converge and diverge simultaneously. A series can either converge or diverge, but not both. This is because the definition of convergence and divergence are mutually exclusive.

What are some real-world applications of understanding the divergence of complex series?

Understanding the divergence of complex series is essential in many fields of science and engineering, including physics, chemistry, and electrical engineering. For example, it is used to analyze the behavior of electrical circuits and to determine the stability of physical systems.

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