1. If you still find yourself still arguing about how the general term $$a_n$$ is going to zero, instead of the nature of the general term, like:

• the general term is exactly the form of some known series (geometric and p-series)
• the general term is a difference (telescoping series)
• the general term is a positive decreasing function of n, that is integrable (integral test)
• the general term is alternating with absolute value decreasing to zero (alternating series test)
• the absolute value ratios of consecutive general terms is converging (ratio test, which apples to alternating series too)

Then be warned that you’re doing something fundamentally wrong. Let me repeat again,

$$\lim_{n\to\infty} a_n = 0$$

says nothing conclusive whatsoever for your series under investigation (the only “exception”: unless you couple it with the knowledge that the series is alternating and absolute values of terms decreasing). Arguing the general term to somehow converge to 0 and quoting “whatever test”, all these amounts to a zero mark in a harsh marking scheme. You know I am harsh against irrelevant details.

It is only when the general term DOESN’T converge to zero that the series must diverge by the Divergence Test (Or Term Test if you read sources other than the textbook). In EVERY case when the series does converge, all efforts into arguing that the general term sequence is going to zero (in whatever vaguely described ways) are going to be in vain. You need to make explicit comparisons or explain the nature of the general term as above to get credit.

2. The comparison tests require your knowledge of another series $$\sum_{n=1}^{\infty} b_n$$ which you know to converge / diverge. If you guess divergence, get a series below: $$b_n \leq a_n$$, if you guess convergence, get a series above. The required conditions for the direct comparison test is that the sequences are positive and the series you use for comparison is known to diverge / converge. The limit comparison test is easier to use, you get another series so that the ratio of terms converges to some nonzero number:

$$\lim \frac{a_n}{b_n} = L \neq 0$$

and of course not infinity (i.e. it diverges). Then either both converge or both diverge.

Posted in Quiz & Midterm