Interval of Convergence for Infinite Series with Ratio Test

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SUMMARY

The discussion centers on finding the Interval of Convergence for the infinite series represented by the summation sigma[n=0,inf] (n (x-2)^n)/( (n+1)4^n ). The Ratio Test was applied, leading to the limit expression [x/4 - 1/2] lim n->inf (n+1)^2 / ((n^2 + 2n) ). The limit equated to 1, which raised questions about the applicability of the Ratio Test. Ultimately, the conclusion reached is that despite the limit being 1, it is acceptable to solve for x under the assumption that x/4 - 1/2 < 1.

PREREQUISITES
  • Understanding of infinite series and convergence
  • Familiarity with the Ratio Test for convergence
  • Basic algebraic manipulation skills
  • Knowledge of limits and their properties
NEXT STEPS
  • Study the application of the Ratio Test in greater detail
  • Learn about the conditions under which the Ratio Test is inconclusive
  • Explore alternative convergence tests such as the Root Test
  • Investigate endpoint behavior in the context of series convergence
USEFUL FOR

Students studying calculus, particularly those focusing on series and convergence, as well as educators seeking to clarify the application of the Ratio Test in determining intervals of convergence.

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Homework Statement



Find the Interval of Convergence for the given series. Check the endpoint behavior carefully sigma[n=0,inf] (n (x-2)^n)/( (n+1)4^n )

Homework Equations





The Attempt at a Solution



I was following along with the answer key and they used the ratio test...
The only problem is that
[x/4 - 1/2] lim n->inf (n+1)^2 / ((n^2 + 2n) ) = x/4 - 1/2
because the lim n->inf (n+1)^2 / ((n^2 + 2n) )
I thought the ratio test was inconclusive if when you took the limit you got 1?
The answer key than proceeded to solve for x assuming
x/4 - 1/2 < 1
can this be done even though the limit equals one?
 
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My bad... I get it now sorry guys...
 

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