Evaluate the integral using integration by parts with the indicated choices of $u$ and $dv.($Use $C$ for the constant of integration.$)$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}xcos(5x)dx$;$,u=x,dv=cos(5x)dx$

Consider the following series.

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{3+e^{-n}}$

What does the Test for Divergence reveal about this series?

Converges; the limit of the terms $a_n$ is a constant as $n$ goes to $.$

Converges; the series is a constant multiple of a geometric series.

Diverges; the limit of the terms $a_n$ is not $0$ as $n$ goes to $.$

Diverges; the sequence $a_n$ diverges as $n$ goes to $.$

Diverges; the series is a constant multiple of the harmonic series.

If it is convergent, find its sum. $($If the quantity diverges, enter DIVERGES.$)$

DIVERGES

Consider the following series.

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{3+e^{-n}}$

What does the Test for Divergence reveal about this series?

Converges; the limit of the terms $a_n$ is a constant as $n$ goes to $.$

Converges; the series is a constant multiple of a geometric series.

Diverges; the limit of the terms $a_n$ is not $0$ as $n$ goes to $.$

Diverges; the sequence $a_n$ diverges as $n$ goes to $.$

Diverges; the series is a constant multiple of the harmonic series.

If it is convergent, find its sum. $($If the quantity diverges, enter DIVERGES.$)$

DIVERGES