Solutions - 0190-01

(a)

$$\begin{array}{c}\left|\frac{{(-1)}^{n-1}}{n^{1/3}}\right|\gg \gg \frac{1}{n^{1/3}}\\ \\ \sum \frac{1}{n^{1/3}}\text{diverges: }p=1/3<1\end{array}$$

Therefore it is not absolutely converging. Proceed to the AST.

• Passes SDT? Yes, $1/n^{1/3}\to 0$.
• Decreasing? Yes, denominator is increasing.

Therefore the AST applies and says this converges. So it converges conditionally.

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(b)

$$\left|\frac{{(-1)}^n{n}^4}{n^3+1}\right|\gg \gg \frac{n^4}{n^3+1}\to \infty$$

So this fails the SDT, hence it diverges.