question 11 of 13 > write an inequality comparing $\frac{n^{2}-6}{n^{5}+6}$ with $\frac{1}{n^{3}}$ for…

question 11 of 13 > write an inequality comparing $\frac{n^{2}-6}{n^{5}+6}$ with $\frac{1}{n^{3}}$ for $ngeq1$. (express numbers in exact form. use symbolic notation and fractions where needed.) inequality: use this inequality to draw a conclusion about the series $sum_{n = 1}^{infty}\frac{n^{2}-6}{n^{5}+6}$. the series diverges by the direct comparison test because $sum_{n = 1}^{infty}\frac{1}{n^{3}}$ also diverges. it is not possible to draw a conclusion about the convergence of the series because $sum_{n = 1}^{infty}\frac{1}{n^{3}}$ diverges. the series converges by the direct comparison test because $sum_{n = 1}^{infty}\frac{1}{n^{3}}$ also converges.

question 11 of 13 > write an inequality comparing $\frac{n^{2}-6}{n^{5}+6}$ with $\frac{1}{n^{3}}$ for $ngeq1$. (express numbers in exact form. use symbolic notation and fractions where needed.) inequality: use this inequality to draw a conclusion about the series $sum_{n = 1}^{infty}\frac{n^{2}-6}{n^{5}+6}$. the series diverges by the direct comparison test because $sum_{n = 1}^{infty}\frac{1}{n^{3}}$ also diverges. it is not possible to draw a conclusion about the convergence of the series because $sum_{n = 1}^{infty}\frac{1}{n^{3}}$ diverges. the series converges by the direct comparison test because $sum_{n = 1}^{infty}\frac{1}{n^{3}}$ also converges.

Answer

Explanation:

Step1: Cross - multiply to compare

For (n\geq1), we want to compare (\frac{n^{2}-6}{n^{5}+6}) and (\frac{1}{n^{3}}). Cross - multiply: ((n^{2}-6)n^{3}) and (n^{5}+6). ((n^{2}-6)n^{3}=n^{5}-6n^{3}).

Step2: Analyze the difference

Find the difference between (n^{5}+6) and (n^{5}-6n^{3}): ((n^{5}+6)-(n^{5}-6n^{3}) = 6 + 6n^{3}>0) for (n\geq1). So (n^{5}+6>(n^{2}-6)n^{3}). Then (\frac{n^{2}-6}{n^{5}+6}<\frac{1}{n^{3}}) for (n\geq1).

Step3: Use the Direct Comparison Test

The (p) - series (\sum_{n = 1}^{\infty}\frac{1}{n^{p}}) diverges when (p = 3>1). Since (0<\frac{n^{2}-6}{n^{5}+6}<\frac{1}{n^{3}}) for (n\geq1) and (\sum_{n=1}^{\infty}\frac{1}{n^{3}}) diverges, we cannot draw a conclusion about the convergence of (\sum_{n = 1}^{\infty}\frac{n^{2}-6}{n^{5}+6}) using the Direct Comparison Test in this case.

Answer:

inequality: (\frac{n^{2}-6}{n^{5}+6}<\frac{1}{n^{3}}) It is not possible to draw a conclusion about the convergence of the series because (\sum_{n = 1}^{\infty}\frac{1}{n^{3}}) diverges.