text: # How do you determine if the series the converges conditionally, absolutely or diverges given Sigma ((-1)^(n+1))/(n^1.5) from [1,oo)?
  
  Jan 12, 2017
  
  The series:
  
  ${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n + 1} / {n}^{\frac{3}{2}}$
  
  is absolutely convergent
  
  #### Explanation:
  
  Given the series:
  
  (1) ${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n + 1} / {n}^{\frac{3}{2}}$
  
  we can test for convergence the series:
  
  (2) ${\sum}_{n = 1}^{\infty} \left\mid {\left(- 1\right)}^{n + 1} / {n}^{\frac{3}{2}} \right\mid = {\sum}_{n = 1}^{\infty} \frac{1}{n} ^ \left(\frac{3}{2}\right)$
  
  If this series converges, then the series (1) converges absolutely ( and also conditionally ).
  
  We can apply the integral test to the series (2) using the function:
  
  $f \left(x\right) = \frac{1}{x} ^ \left(\frac{3}{2}\right)$
  
  As, for $x \in \left[1 , + \infty\right)$, $f \left(x\right)$ is positive and decreasing, it is infinitesimal for $x \to + \infty$ and $f \left(n\right) = \frac{1}{n} ^ \left(\frac{3}{2}\right)$.
  
  Se we calculate:
  
  ${\int}_{1}^{\infty} \frac{\mathrm{dx}}{x} ^ \left(\frac{3}{2}\right) = {\left[- \frac{2}{x} ^ \left(\frac{1}{2}\right)\right]}_{1}^{\infty} = 2$
  
  which means that the series (1) is absolutely convergent.
explanation: The document contains a math question and a detailed answer. The question asks how to determine if a given series converges conditionally, absolutely, or diverges. The answer provides a step-by-step explanation using the integral test to show that the series is absolutely convergent.
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distilabel_metadata.content: # How do you determine if the series the converges conditionally, absolutely or diverges given Sigma ((-1)^(n+1))/(n^1.5) from [1,oo)?
  
  Jan 12, 2017
  
  The series:
  
  ${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n + 1} / {n}^{\frac{3}{2}}$
  
  is absolutely convergent
  
  #### Explanation:
  
  Given the series:
  
  (1) ${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n + 1} / {n}^{\frac{3}{2}}$
  
  we can test for convergence the series:
  
  (2) ${\sum}_{n = 1}^{\infty} \left\mid {\left(- 1\right)}^{n + 1} / {n}^{\frac{3}{2}} \right\mid = {\sum}_{n = 1}^{\infty} \frac{1}{n} ^ \left(\frac{3}{2}\right)$
  
  If this series converges, then the series (1) converges absolutely ( and also conditionally ).
  
  We can apply the integral test to the series (2) using the function:
  
  $f \left(x\right) = \frac{1}{x} ^ \left(\frac{3}{2}\right)$
  
  As, for $x \in \left[1 , + \infty\right)$, $f \left(x\right)$ is positive and decreasing, it is infinitesimal for $x \to + \infty$ and $f \left(n\right) = \frac{1}{n} ^ \left(\frac{3}{2}\right)$.
  
  Se we calculate:
  
  ${\int}_{1}^{\infty} \frac{\mathrm{dx}}{x} ^ \left(\frac{3}{2}\right) = {\left[- \frac{2}{x} ^ \left(\frac{1}{2}\right)\right]}_{1}^{\infty} = 2$
  
  which means that the series (1) is absolutely convergent.
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 how to determine if a given series converges conditionally, absolutely, or diverges. The answer provides a step-by-step explanation using the integral test to show that the series is absolutely convergent.
  
  Label: question-and-answer