Consider the series.

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{5n}{(n^2+1{)}^{\frac{3}{5}}}$

Let $f(x)=\frac{5x}{(x^2+1{)}^{\frac{3}{5}}}.$ Find $f^{\text{'}}(x).$

$(Express$ numbers $in$ exact form. Use symbolic notation and fractions where needed.$)$

$f^{\text{'}}(x)$

Evaluate ${\displaystyle \underset{3}{\overset{}{}}}\frac{5x}{(x^2+1{)}^{\frac{3}{5}}}dx.$

$(Express$ numbers $in$ exact form. Use symbolic notation and fractions where needed.$)$

${\displaystyle \underset{3}{\overset{}{}}}\frac{5x}{(x^2+1{)}^{\frac{3}{5}}}dx=$

Identify the true statement$(s)$ about the series and the integral.

The Integral Test does not apply because $fis$ not continuous $on(1,).$

The Integral Test does not apply because $fis$ only increasing when $x1.$

$f^{\text{'}}(x)<mn>0</mn>$ for all $x3,sofis$ eventually decreasing.

The integral ${\displaystyle \underset{3}{\overset{}{}}}f(x)dxis$ finite, $so$ the series converges $by$ the Integral Test.

The integral ${\displaystyle \underset{3}{\overset{}{}}}f(x)dxis$ infinite, $so$ the series diverges $by$ the Integral Test.

$f(x)>0$ and $is$ continuous for all $x3.$