To determine whether the series with the general term $\frac{n+1}{(n+2)^3}$ is convergent or divergent, we can use several methods, such as the comparison test, the ratio test, or the integral test. Here, we will use the comparison test.

Comparison Test

We can compare $\frac{n+1}{(n+2)^3}$ with a simpler term that we know converges or diverges. Notice that for large $n$, the term $\frac{n+1}{(n+2)^3}$ behaves similarly to $\frac{n}{n^3} = \frac{1}{n^2}$.

Let's compare $\frac{n+1}{(n+2)^3}$ with $\frac{1}{n^2}$:

$\frac{n+1}{(n+2)^3} \approx \frac{n}{n^3} = \frac{1}{n^2} \text{ as } n \to \infty$

We know that the series $\sum \frac{1}{n^2}$ is a convergent $p$-series, with $p = 2$ (since $p > 1$).

Formal Comparison

To be rigorous, we will show that $\frac{n+1}{(n+2)^3}$ is less than a constant multiple of $\frac{1}{n^2}$ for sufficiently large $n$:

$\frac{n+1}{(n+2)^3} < \frac{n+1}{n^3} \text{ for large } n$

We can simplify this further:

$\frac{n+1}{n^3} = \frac{1}{n^2} + \frac{1}{n^3}$

Since $\frac{1}{n^3}$ is positive and decreases faster than $\frac{1}{n^2}$, the term $\frac{n+1}{n^3}$ is certainly less than $\frac{2}{n^2}$ for large $n$:

$\frac{1}{n^2} < \frac{n+1}{n^3} < \frac{2}{n^2}$

Conclusion

Since $\frac{n+1}{(n+2)^3}$ is less than a constant multiple of $\frac{1}{n^2}$ and the series $\sum \frac{1}{n^2}$ converges, by the comparison test, the series $\sum \frac{n+1}{(n+2)^3}$ also converges.

Would you like more details or have any questions about this explanation?

Here are some related questions:

1. What is the comparison test in series convergence?
2. How do you determine if a $p$-series converges or diverges?
3. What is the integral test for series convergence?
4. How does the ratio test help determine series convergence?
5. Can the root test be used for this series?
6. What is the limit comparison test?
7. How do you prove a series is absolutely convergent?
8. What happens to a series if the general term does not approach zero?

Tip: When using the comparison test, it is helpful to compare the given series to a known benchmark series, such as a $p$-series, where the behavior is well understood.