Chapter 10
Polynomial and Series Representations of Functions





10.5 Series of Constants

In this section we study some of the specific values of functions defined by infinite series, such as the Taylor series for `ln text[(] 1+x text[)]` and for `text(arctan) x`. We also begin our study of whether a given series actually has a value — that is, whether its terms get small fast enough that infinitely many terms can be added to get a finite answer. Some aspects of this question are easy. For example, we already know that the geometric series, which represents `1 text[/] text[(] 1-x text[)]`, has a finite sum if `-1 < x < 1`, but not otherwise. Here we will see the first instance in which the question becomes subtle: a series whose terms get small, but not fast enough to have a finite sum.

10.5.1 Convergence and Divergence

We start with a less subtle example, a series whose terms do not get small. For your convenience, we repeat the formula for the logarithmic series here:

ln ( 1 + x ) = x - x 2 2 + x 3 3 - x 4 4 + ⋅⋅⋅ = ∑ k = 0 ∞ ( - 1 ) k x k + 1 k + 1 , if `text[|] x text[|] < 1.`

The subject of the next activity is the result of substituting `x = 2` in this series:

2 - 2 2 2 + 2 3 3 - 2 4 4 + ⋅⋅⋅ = ∑ k = 0 ∞ ( - 1 ) k 2 k + 1 k + 1 .

We will see that this series does not add up to `text(ln) 3` — or to anything else, for that matter.

Activity 1

  1. Use your graphing tool to sketch the graph of `f text[(] x text[)] = 2^xtext[/]x` for `x > 0`. Describe the behavior of `f text[(] x text[)]` as `x rarr oo`.

  2. Explain why `2^m//m rarr oo` as `m rarr oo`. (Be careful: Both numerator and denominator are approaching `oo`.)

  3. Why does this behavior of `2^m//m` imply that the infinite series

    ∑ k = 0 ∞ ( - 1 ) k 2 k + 1 k + 1

    does not converge to anything? What are the partial sums doing as the number of terms increases? Generate the first `10` partial sums to confirm your conclusion.

  4. In general, if an infinite series

    ∑ k = 0 ∞ b k

    converges, what can you say about

    lim k → ∞ b k ?

Comment 1Comment on Activity 1

We state the condition just noted in the Comment on Activity 1 as a formal test for divergence.

Divergence Test   If the sequence `b_0,b_1,b_2,...` does not converge to `0`, then
∑ k = 0 ∞ b k
diverges.

Notice that we use the symbolism

∑ k = 0 ∞ b k

to mean two different things. On the one hand, it designates an infinite series — a sum with infinitely many terms — which may or may not converge to something. On the other hand, if the series converges, this notation represents the actual sum of the series — a number. This is not the first time you have seen one notation standing for two different concepts, nor will it be the last.

We know that the logarithmic series converges for `-1 < x < 1`, and the Divergence Test tells us that it diverges for `text[|] x text[|] > 1`. We turn our attention now to the two interesting series that result from substituting `x = 1` and `x = -1` into the logarithmic series. These numbers are the endpoints of the interval of convergence. Depending on what we find out about the series that result from these substitutions, an endpoint may or may not be in the interval of convergence. That is, the resulting series of constants may or may not converge.

Here is the series we get by substituting `x = 1` into the Taylor series for `text(ln) text[(] 1 + x text[)]`:

1 - 1 2 + 1 3 - 1 4 + ⋅⋅⋅ + ( - 1 ) k k + 1 + ⋅⋅⋅ = ∑ k = 0 ∞ ( - 1 ) k k + 1 .

Notice in particular the following features of this series:

The series in Activity 1 shares the first of these properties but not the second.

Definition   A series is called alternating if its terms strictly alternate in sign. It does not matter whether the alternating sign pattern starts with positive or negative.

The alternating series we obtained by substituting `x = -1` into the Taylor series for `text(ln) text[(] 1 + x text[)]` has a special name.

Definition   The alternating harmonic series is the infinite series
1 - 1 2 + 1 3 - 1 4 + ⋅⋅⋅ ⁢   .

When we substitute `x = -1` in the expression `text[(]-1text[)]^kx^(k+1)`, we get `text[(]-1text[)]^(2k+1).` Because the power is odd for every `k`, this expression is always `-1`. Thus the series that results from substituting `-1` in the logarithmic series is

- 1 - 1 2 - 1 3 - 1 4 - ... ⁢   - 1 k + 1 - ... = ∑ k = 0 ∞ - 1 k + 1 .
Definition   A series is called monotonic if its terms all have the same sign. It does not matter whether the constant sign is positive or negative.

The series

- 1 - 1 2 - 1 3 - 1 4 - ⋅⋅⋅ ⁢   - 1 k + 1 - ⋅⋅⋅

looks like a close relative of the alternating harmonic series. The corresponding series with all plus signs is one of the most important of all infinite series.

Definition   The harmonic series is the infinite series
1 + 1 2 + 1 3 + 1 4 + ⋅⋅⋅ ⁢       .

Thus the second endpoint series we get from the logarithmic series is the negative of the harmonic series. Both the harmonic series and its negative are monotonic. Both have terms that approach zero. But so far all we know about either the harmonic series or the alternating harmonic series is a name. Do these series converge or diverge? If either converges, to what number does it converge? In spite of the similarity of the two series, these questions have to be answered in quite different ways for the two cases. We will study the harmonic series and the alternating harmonic series separately in the next two subsections.

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