Which one $is$ true about the series ${\displaystyle \underset{n=1}{\overset{}{}}}(\frac{4n-1}{n^4+1}{)}^{\frac{1}{2}}?$

$(a)$ The series $is$ convergent $bylimit$ comparison test.

$(b)$ The series $is$ convergent $by$ the root test.

$(c)$ The series $is$ divergent $by$ direct comparison test.

$(d)$ The series $is$ divergent $by$ the ratio test.

$(e)$ The series $is$ divergent $by$ the root test.

Which one $is$ true about the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^3+1}?$

$(a)$ The series $is$ divergent $by$ the root test.

$(b)$ The series $is$ convergent $by$ the root test.

$(c)$ The series $is$ convergent $by$ direct comparison test comparing with the $p-$series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{\sqrt[3]{n}}.$

$(d)$ The series $is$ convergent $by$ direct comparison test comparing with the $p-$series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^3}.$

$(e)$ The series $is$ divergent $by{n}^{th}-$term test.

Which one $is$ true about the series ${\displaystyle \underset{n=1}{\overset{}{}}}(-1{)}^{n+1}\frac{1}{\sqrt[\phantom{2}]{n+1}}?$

$(a)$ The series $isabsolutely$ convergent $by$ ratio test.

$(b)$ The series $is$ not $absolutely$ convergent, but conditionally convergent $by$ the alternating series test.

$(c)$ The series $is$ convergent $by$ direct comparison test comparing with the $p-$series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{\sqrt[\phantom{2}]{n}}.$

$(d)$ The series $isabsolutely$ convergent $by$ root test.

$(e)$ The series $is$ convergent $by{n}^{th}-$term test.