Hello! I'm not entirely sure which methods you have been taught to answer this type of question (I was a Science student and not an Engineering one), but I'll try to provide reasoning in words along with the more formal approaches:
At first glance, it looks like both (a) and (d) should be divergent series. My 'gut instinct' tells me that for (a), as n increases, n^3 - 1 and n^3 + 1 both approach n^3, and so the value of the term being summed should approach 1; therefore beyond a certain point the series almost becomes . .. + 1 + 1 + 1 + 1... , which would make it divergent. For a more formal approach: rewrite (n^3 -1)/(n^3 + 1) as 1 - 2/(n^3 + 1). Then, as n approaches positive infinity, -2/(n^3 + 1) approaches 0 from the negative side. Therefore, each term being summed approaches 1, and hence the series diverges.
For d), my gut instinct tells me that 2^n should grow much more quickly than n^2 does, and hence we have an even more extreme case than in a): the individual terms should in fact grow to infinity, which would of course make the series divergent. A good rule of thumb I like to use is that factorial series outgrow power series, which outgrow polynomial series. How I would do this question formally: I would try the ratio test. If my working out is correct, dividing the (k+1)th term in the series by the kth term should give (2k^2)/(k+1)^2. This can be rewritten as 2(1 - 1/(k+1))^2. As k approaches infinity, this approaches 2. Therefore, by the ratio test, since the ratio of successive terms is > 1, the series must be divergent.
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