Find the radius of convergence

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SUMMARY

The radius of convergence for the series \(\sum_{n=1}^{\infty}\frac{(2n)!x^n}{(n!)^2}\) is determined to be \(1/4\). The Ratio Test was applied, leading to the expression \(L = |\frac{2x}{n+1}|\). A common mistake identified was the incorrect application of factorials, specifically in the expansion of \((2(n+1))!\), which should be expressed as \((2n + 2)(2n + 1)(2n)!\) rather than \((2n + 2)(2n)!\).

PREREQUISITES
  • Understanding of series convergence and divergence
  • Familiarity with the Ratio Test for series
  • Knowledge of factorial notation and properties
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the application of the Ratio Test in more complex series
  • Learn about other convergence tests such as the Root Test and Comparison Test
  • Explore advanced topics in combinatorial mathematics related to factorials
  • Investigate the implications of radius of convergence in power series
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Students studying calculus, mathematicians focusing on series and convergence, and educators teaching advanced algebra concepts.

JulioMarcos
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Homework Statement
Find the radius of convergence of the Series:
[tex]\sum_{i=1}^{\infty}\frac{(2n)!x^n}{(n!)^2}[/tex]

The attempt at a solution
I used the Ratio Test but I always get L = [tex]|\frac{2x}{n+1}|[/tex]

The answer is 1/4. I think I am mistaking with factorial.
 
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Solved. The problem was tha
(2(n+1))! = (2n + 2)(2n + 1)(2n)!
and I was doing (2(n+1))! = (2n + 2)(2n)!
because i thought you should distribute the 2
 
Ah it is always good to find your own mistake. Off the bat that was my guess... Sadly I got in the thread too late.
 

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