text: Student Question
  
  # `sum_(n=1)^oo (-1)^(n-1)(3/2)^n/n^2` Use the Root Test to determine the convergence or divergence of the series.
  
  In using Root test on a series sum a_n, we determine the limit as:
  
  `lim_(n-gtoo) root(n)(|a_n|)= L`
  
  or
  
  `lim_(n-gtoo) |a_n|^(1/n)= L`
  
  a) `Llt1` then the series is absolutely convergent.
  
  b) `Lgt1` then the series is divergent.
  
  c) `L=1` or does not exist  then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.
  
  We may apply the Root Test to determine the convergence or divergence of the series `sum_(n=1)^oo (-1)^(n-1) *(3/2)^n/n^2` .
  
  `lim_(n-gtoo) |((-1)^(n-1) *(3/2)^n/n^2 )^(1/n)| =lim_(n-gtoo) |(-1)^((n-1)*1/n)(3/2)^(n*1/n)/n^(2*1/n)|`
  
  `=lim_(n-gtoo) |(-1)^(n/n-1/n)(3/2)^(n/n)/n^(2/n)|`
  
  `=lim_(n-gtoo)( 1 * (3/2)^1/n^(2/n))`
  
  `=lim_(n-gtoo) (3/2)/n^(2/n)`
  
  Note: `|(-1)^(n/n-1/n)| = 1`
  
  Apply the limit property: `lim_(x-gta)[(f(x))/(g(x))] =(lim_(x-gta) f(x))/(lim_(x-gta) g(x)).`
  
  `lim_(n-gtoo) (3/2)/n^(2/n)=(lim_(n-gtoo) 3/2)/(lim_(n-gtoo)n^(2/n))`
  
  `= ((3/2))/1`
  
  ` =3/2 or 1.5`
  
  The limit value `L = 3/2 or 1.5` satisfies the condition: `Lgt1` since `3/2gt 1` or `1.5gt1` .
  
  Thus, the series `sum_(n=1)^oo (-1)^(n-1) *(3/2)^n/n^2` is divergent
explanation: The document contains a math question and a detailed answer. The question asks about using the Root Test to determine the convergence or divergence of a specific series. The answer provides a step-by-step application of the Root Test and concludes that the series is divergent.
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distilabel_metadata.content: Student Question
  
  # `sum_(n=1)^oo (-1)^(n-1)(3/2)^n/n^2` Use the Root Test to determine the convergence or divergence of the series.
  
  In using Root test on a series sum a_n, we determine the limit as:
  
  `lim_(n-gtoo) root(n)(|a_n|)= L`
  
  or
  
  `lim_(n-gtoo) |a_n|^(1/n)= L`
  
  a) `Llt1` then the series is absolutely convergent.
  
  b) `Lgt1` then the series is divergent.
  
  c) `L=1` or does not exist  then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.
  
  We may apply the Root Test to determine the convergence or divergence of the series `sum_(n=1)^oo (-1)^(n-1) *(3/2)^n/n^2` .
  
  `lim_(n-gtoo) |((-1)^(n-1) *(3/2)^n/n^2 )^(1/n)| =lim_(n-gtoo) |(-1)^((n-1)*1/n)(3/2)^(n*1/n)/n^(2*1/n)|`
  
  `=lim_(n-gtoo) |(-1)^(n/n-1/n)(3/2)^(n/n)/n^(2/n)|`
  
  `=lim_(n-gtoo)( 1 * (3/2)^1/n^(2/n))`
  
  `=lim_(n-gtoo) (3/2)/n^(2/n)`
  
  Note: `|(-1)^(n/n-1/n)| = 1`
  
  Apply the limit property: `lim_(x-gta)[(f(x))/(g(x))] =(lim_(x-gta) f(x))/(lim_(x-gta) g(x)).`
  
  `lim_(n-gtoo) (3/2)/n^(2/n)=(lim_(n-gtoo) 3/2)/(lim_(n-gtoo)n^(2/n))`
  
  `= ((3/2))/1`
  
  ` =3/2 or 1.5`
  
  The limit value `L = 3/2 or 1.5` satisfies the condition: `Lgt1` since `3/2gt 1` or `1.5gt1` .
  
  Thus, the series `sum_(n=1)^oo (-1)^(n-1) *(3/2)^n/n^2` is divergent
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer. The question asks about using the Root Test to determine the convergence or divergence of a specific series. The answer provides a step-by-step application of the Root Test and concludes that the series is divergent.
  
  Label: question-and-answer