10.4.1 Geometric Sums and Series

Although you may not have realized it back in Section 10.1 and Section 10.3, when we were discussing formulas for finite and infinite geometric sums, we were really investigating polynomial approximation of functions. We begin this section by reviewing, from this perspective, what we know about geometric sums.

If `r` is any number other than `1`, and `n` is any nonnegative integer, then

If `text[|] r text[|] < 1`, then,

so

This says that the function

is approached by the geometric polynomials

as `n` approaches infinity, at least for values of `text[|] x text[|]` with `text[|] x text[|] < 1`. In the notation of infinite series,

Representation of this function by approximating polynomials is not especially important because its values can be computed by simple arithmetic from the formula `f text[(] x text[)] = 1text[/(]1-xtext[)]` and without any artificial restriction on the size of `x`. Indeed, when the subject of geometric series first came up, our problem was the infinitely long sum, and its solution was to find the simple quotient expression — not the other way around. However, we will soon see that this function is intimately related to other important functions, especially the natural logarithm and inverse tangent functions. The relationships carry over to the approximating polynomials, and that enables us to find polynomial approximations to these other functions from the geometric polynomials, `1 + x + x^2 + cdots + x^n`.

Activity 1

1. Find the Taylor polynomials of degrees `0` through `5` for the function `f text[(] x text[)] = 1text[/(]1-xtext[)]`. How do these polynomials compare to the geometric polynomials?

2. In a single graphing window, graph `f text[(] x text[)] ` and the six Taylor polynomials from part (a).

3. Evaluate the Taylor polynomials in part (a) at each of the following values of `x`: `0.9, -0.9, 1, -1, 1.1, -1.1`. For each choice of `x`, describe in words how the polynomial value at `x` compares with `f text[(] x text[)] .`

Comment on Activity 1

For the function `f text[(] x text[)] = 1text[/(]1-xtext[)]`, we find that the Taylor polynomials do not converge to the function for all `x`, but only for a limited range of `x`'s: `-1 < x < 1`. In fact, for `x ≥ 1`, we see that the values of `1 + x + x^2 + cdots + x^n` grow without bound as `n` becomes large. Similarly, for `x < -1`, the successive polynomial values alternate in sign and eventually become large in absolute value. And for the special value `x = -1`, the values alternate between `0` and `1`, never settling down to a limiting value.