级数和极限对比测试是什么?

Suppose $a_{1} \leq b_{1}$ and $a_{2} \leq b_{2}$. Then $a_{1} + a_{2} \leq b_{1} + b_{2}$ . This trivial observation turns out to be the root of an incredibly powerful method for testing the convergence or divergence of infinite series.

An obvious generalization of the above is that if we have two infinite sequences $a_{n}, b_{n}$ satisfying $a_{n} \leq b_{n}$ for all $n$, then $a_{1} + \ldots + a_{n} < b_{1} + \ldots + b_{n}$ for all $n$. Suppose in addition that $0 \leq a_{n}, b_{n}$ for all $n$, and that $\sum b_{n}$ converges to its sum $L$. Then $0 \leq a_{1} + \ldots + a_{n} \leq b_{1} + \ldots + b_{n} \leq L$. However, the partial sums for $\sum a_{n}$ are non-negative and bounded from above. Therefore, by an earlier theorem (this probably needs to be filled out in an earlier topic!), $\sum a_{n}$ converges as well. Intuitively, if $b_{n}$ is "small enough" such that $\sum b_{n}$ converges, and $a_{n}$ is smaller than $b_{n}$, then $a_{n}$ is also "small enough" so that $\sum a_{n}$ converges as well.

Similarly, if $0 \leq a_{n} \leq b_{n}$, but this time $\sum a_{n}$ diverges, then $\sum b_{n}$ also diverges as well. Indeed, $\sum b_{n}$ must diverge, because if it converged, then so would $\sum a_{n}$ by what we have just shown, a contradiction. Therefore, we have the following theorem:

Theorem (Direct Comparison Test). Suppose $0 \leq a_{n} \leq b_{n}$ for all $n$.

1. If $\sum b_{n}$ converges, then so does $\sum a_{n}$.

2. If $\sum a_{n}$ diverges, then so does $\sum b_{n}$.

Exercise 1. Show that the requirement that $a_{n}, b_{n}$ be non-negative in the direct comparison test is essential. That is, exhibit a convergent series $\sum b_{n}$ such that $a_{n} < b_{n}$ but $\sum a_{n}$ diverges.

Exercise 2. Determine the convergence or divergence of the series

$$\sum_{n=1}^{\infty} \frac{1}{n^2 e^{n}}.$$

Notice that if we want to use this test to show that a particular series converges or diverges, we need to be able to compare that series to some reference series whose convergence or divergence we already know. And what general series do we know converge or diverge? Geometric series! Throughout the remainder of the sections on infinite series, we will keep returning to comparisons against geometric series, or theorems which are proven via comparison against geometric series, over and over.

(There's more stuff to write here: actual concrete examples/exercises, and the limit comparison test. I'll do it later.)