Answer
$$
\sum_{n=2}^{\infty} \frac{2+\sin n}{ n}
$$
Apply the Integral Test to the series
$$
\sum_{n=2}^{\infty} \frac{2+\sin n}{ n}
$$
The function
$$
f(x)=\frac{2+\sin x}{ x}
$$
is positive and continuous for $x\geq 1$ . To determine whether $f$ is decreasing or not, find the derivative.
$$
f^{'}(x)=\frac{x \cos x - (2+ \sin x)}{ x^{2}}
$$
we obtain that, $f^{'}(x) \gt 0 $ for some $x\gt1$,
so $ f $ is not decreasing for some $x\gt1$,
and it follows that $f$ does not satisfy the conditions for the Integral Test.
Work Step by Step
$$
\sum_{n=2}^{\infty} \frac{2+\sin n}{ n}
$$
Apply the Integral Test to the series
$$
\sum_{n=2}^{\infty} \frac{2+\sin n}{ n}
$$
The function
$$
f(x)=\frac{2+\sin x}{ x}
$$
is positive and continuous for $x\geq 1$ . To determine whether $f$ is decreasing or not, find the derivative.
$$
f^{'}(x)=\frac{x \cos x - (2+ \sin x)}{ x^{2}}
$$
we find that, $f^{'}(x) \gt 0 $ for some $x\gt1$,
so $ f $ is not decreasing for $x\gt1$,
and it follows that $f$ does not satisfy the conditions for the Integral Test.
