Final answer:
The series given by the sum from k=1 to infinity of (1 + 3^k) / (1 + 2^k) diverges. This is determined by comparing it to a divergent geometric series and the behavior of the dominant terms as k approaches infinity.
Step-by-step explanation:
The question asks whether the series given by the sum from k=1 to infinity of (1 + 3k) / (1 + 2k) converges. To determine this, we can use the comparison test for series convergence by comparing our series to another that has a known behavior.
Let's compare each term of our series to 3k/2k, simplifying to (3/2)k. Because (3/2)k is a geometric series with a ratio greater than 1, it diverges. Since each term of our original series is eventually larger than the corresponding term of (3/2)k, it follows that our original series also diverges.
This conclusion can also be supported by considering the terms of the series as k approaches infinity. The dominant term in the numerator and denominator is 3k and 2k respectively, and since the higher power of 3 grows faster than the power of 2, the terms do not tend to zero, indicating divergence.