This is an infinite series given by:

$\sum_{n=1}^\infty \frac{2 + n}{1 - 3n}.$

Steps to Solve or Analyze Convergence

1. General Form: The term of the series is: $a_n = \frac{2 + n}{1 - 3n}.$

2. Check Convergence:

   • To determine convergence, analyze the behavior of the term $a_n$ as $n \to \infty$. Specifically, if $a_n \not\to 0$, the series diverges.
   • The dominant term in the numerator is $n$, and in the denominator, it is $-3n$. So, $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{2 + n}{1 - 3n} = \lim_{n \to \infty} \frac{n}{-3n} = \frac{-1}{3}.$

   Since $a_n \to -\frac{1}{3} \neq 0$, the series diverges.

Conclusion

The infinite series $\sum_{n=1}^\infty \frac{2 + n}{1 - 3n}$ diverges because the terms of the sequence do not approach zero.

Would you like further details on series convergence, or do you have additional questions?

Related Questions

1. What are the criteria for a series to converge or diverge?
2. How does the divergence test work for infinite series?
3. Can this series be rearranged or modified to make it converge?
4. What happens if we consider a partial sum of this series?
5. How do we deal with series that contain alternating terms?

Tip: Always verify the limit of $a_n$ before proceeding to more advanced convergence tests.