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You might try exercises #140-145, 152-155, 160-163, 170-171, 175-176, 190, 192 from that section.
We can think of it as a function $a(x)$, but with integers for arguments, and real numbers for values: \[ a: \mathbb{N} \rightarrow \mathbb{R} \]
It might be better simply to write it as $a(n)$, but mathematicians have adopted the alternative notation $a_n$, perhaps to make clear that this function has a domain restricted to integers.
And this is one of the reasons that we are not too surprised when we find ourselves comparing series and improper integrals.
This is completely analogous to the process of checking the convergence of an improper integral, however. I'll describe that process, and include in parentheses the analogue for checking series:
To check the convergence of the integral I (series S) of function $f(x)$ (sequence $a_k$) \[ I=\int_1^\infty f(x)\ dx \] e.g. $f(x)=\frac{1}{x^2}$, we construct function $g(R)$ (sequence $s_n$) \[ g(R)=\int_1^R\frac{1}{x^2}\ dx \]
and check the convergence of function $g(R)$ (sequence $s_n$) as its argument goes to $\infty$ ($R\to\infty$ for the integral, $n\to\infty$ -- by integers only -- for the sequence).
The primary difference is that one process occurs on the real numbers, and the other process occurs on the set of integers we call the natural numbers. To add up -- we might say "accumulate" -- real-valued functions, we use integrals; to add up sequences, we use sums.
So what we've seen so far is that, for series \(\sum_{k=1}^{\infty}a_k\) with terms \(a_k=f(k)\) for some positive, decreasing function with limit \(\lim_{x \to \infty}f(x)=0\), we have the following set of inequalities: \[ \int_{n+1}^\infty f(x)dx \le \sum_{k=n+1}^\infty a_k \le \int_{n}^\infty f(x)dx \le \sum_{k=n}^\infty a_k \] We can think of this as \[ \int_{n+1}^\infty f(x)dx \le LRR(n+1) = RRR(n) \le \int_{n}^\infty f(x)dx \le LRR(n) \] We can use these for either of two purposes:
Because of these bounds, it should be clear that the integral and the sum converge together (since their tails are used to bound each other:
This is a sort of squeeze theorem....
In the introduction to this section our author's say this: We have seen that the integral test allows us to determine the convergence or divergence of a series by comparing it to a related improper integral. In this section, we show how to use comparison tests to determine the convergence or divergence of a series by comparing it to a series whose convergence or divergence is known. Typically these tests are used to determine convergence of series that are similar to geometric series or p-series.
Remember that it's all about the tail. In terms of convergence, we don't care about any handful of terms at the head of a series: \[ \sum_{k=1}^\infty a_k = \sum_{k=1}^n a_k + \sum_{k=n+1}^\infty a_k \] Convergence is about the tail: \(\displaystyle \sum_{k=n+1}^\infty a_k\)