Chapter Review

Our study of normal distributions in Chapter 9 led us to the error function,

erf ( x ) = 2 π ∫ 0 x e - t 2 d t ,

which led to the question of how computers and calculators evaluate such functions. That in turn led us to ask how they evaluate more familiar functions, such as `sin x` and `e^x`. In this chapter we have seen one way to evaluate such functions by approximating polynomials.

We began the chapter with a study of geometric sums, which provide approximating polynomials for a very simple function, `1 text[/]  text[(] 1-x text[)]`, but which play a much more important role later in the chapter. Since the chapter is largely about limiting behavior as a counting index becomes large, we digressed briefly to introduce standard notation for the various limiting behaviors encountered throughout the course.

Next we studied the Taylor polynomials (at `x=0`) for general functions `f.` These polynomials have the form

P n ( x ) = ∑ k = 0 n c k ⁢     x k , where c k = f ( k ) ( 0 ) k ! for k = 0 ,   1 ,   2 ,   ... ,   n .

For many functions, including the exponential, sine, and cosine functions, as `n` gets large, `P_n text[(] x text[)]` approaches `f text[(] x text[)]` for all `x`. For other functions, including `ln x` and `text(arctan) x`, this convergence occurs only for `x`'s in a particular interval.

Our investigation of approximating polynomials led to the concept of a “polynomial of infinite degree” or infinite series. For example, the formula

e x = 1 + x + x 2 2 + ⋅⋅⋅ + x k k ! + ⋅⋅⋅ = ∑ k = 0 ∞     x k k !

contains the same information on approximation of `e^x` as does listing the individual polynomials `1+x+x^2text[/]2+cdots+x^ntext[/]n!`, but in a more succinct form. We combined the Taylor series for `e^x` with substitution and term-by-term integration to find a Taylor series for the error function, which gave us a possible answer to the evaluation question for this function.

We studied several special series that illustrate the subtleties of the notion of convergence. The harmonic series `sum_(k=1)^oo 1/k` does not converge — the partial sums grow without bound, but very slowly. This series diverges because the terms `1//k` do not become small quite fast enough as `k` becomes large. On the other hand, the geometric series `sum_(k=1)^oo 1/2^k` does converge (to `1`). The terms `1text[/]2^k` do become small fast enough to add up infinitely many of them.

A case of special interest is that of alternating series. The prototype is the alternating harmonic series,

1 - 1 2 + 1 3 - 1 4 + 1 5 - ⋅⋅⋅ .

Here we observed that, in addition to alternating in sign, the terms steadily decrease to `0` in absolute value. The Alternating Series Test tells us that series with these properties always converge, and the error in any partial sum is smaller than the absolute value of the first term not used in the partial sum.

Some of our important series representations do not have alternating sign patterns — or do not have them for all numbers in their domains. To estimate error in these cases, we introduced a tool that is not sensitive to sign patterns: comparison of the tail of a series with a geometric series. This tool is harder to apply than the Alternating Series Test, but it resolved all the remaining questions about whether important series did in fact converge.

We consolidated our knowledge of comparison to geometric series in the Ratio Test, a method that is easier to apply but that does not directly provide an error estimate for a partial sum. It does, however, show that all power series have an interval of convergence that is symmetric about the reference point. In fact, it resolves all convergence questions for power series except for what happens at endpoints of the interval of convergence.