Chapter 10
Polynomial and Series Representations of Functions





10.1 Sums and Limits

10.1.1 Geometric Sums

We begin with a simple probability question. What is the probability of tossing a fair coin five times and obtaining at least one head?

Note 1 Note 1 – Discrete probability

We'll begin with the somewhat simpler problem of determining the probability of obtaining at least one head in two tosses. Table 1 shows all the possible outcomes, with "H" representing the toss of a head and "T" representing the toss of a tail.

Table 1   Possible Outcomes of
Two Flips of a Fair Coin
 
Toss 1
Toss 2
Outcome 1
H
H
Outcome 2
H
T
Outcome 3
T
H
Outcome 4
T
T

Since the coin is fair, all outcomes are equally likely. Three of the possible outcomes include at least one head and only one represents no heads. Thus the probability of at least one head is `3` out of `4` or `3//4`. We may write this as `1//2` (the probability of a head on the first toss) plus `1//4` (the probability of a tail on the first toss and a head on the second).

Activity 1

  1. Construct a table of outcomes for three tosses of a fair coin, and calculate the probability that at least one is a head.

  2. Explain why the probability of at least one head in `n` tosses of a fair coin is

    1 2 + 1 4 + ⋅⋅⋅ + 1 2 k + ⋅⋅⋅ + 1 2 n = 1 - 1 2 n = 2 n - 1 2 n .

  3. Find the probability of at least one head in five tosses of a fair coin.

Comment 1Comment on Activity 1

The sums in Activity 1,

S = 1 2 + 1 2 2 + ⋅⋅⋅ + 1 2 n ,

are special cases of geometric sums. More generally, a geometric sum is a sum of the form

S = r m + r m + 1 + ⋅⋅⋅ + r n ,

where `m` and `n` are nonnegative integers and `r` is a real number. Note that each term in the sum (after the first) is obtained from the term to its left by multiplying by `r`.

Geometric sums occur frequently in many parts of mathematics. They are particularly important in this chapter because there is a simple formula for these sums — and, as we shall see, that formula leads to polynomial approximations to an important non-polynomial function.

Example 1

Find a simple formula for

S = r + r 2 + ⋅⋅⋅ + r n ,

where `n` is a positive integer and `r` is a real number.

Solution   If `r = 1`, then `S = n`. If `r != 1`, then

r ⁢ S = r 2 + ⋅⋅⋅ + r n + r n + 1 .

So

S - r ⁢ S = r - r n + 1

or

( 1 - r ) ⁢ S = r - r n + 1 .

Dividing by `1 - r`, we obtain

⁢ S = r - r n + 1 1 - r .

In particular if `r = 1//2`, then

⁢ S ⁢ = 1 2 - 1 2 n 1 - 1 2
  ⁢ = 1 2 - 1 2 n 1 2
  ⁢ = 1 - 1 2 n = 2 n - 1 2 n ,

end solutionas we determined above.

Activity 2

  1. If `n` is a positive integer and `r != 1`, show that

    ⁢ 1 + r + r 2 + ⋅⋅⋅ + r n = 1 - r n + 1 1 - r .

  2. If `m` and `n` are nonnegative integers with `n > m` and `r != 1`, find a formula for the geometric sum

    ⁢ r m + r m + 1 + ⋅⋅⋅ + r n .

Comment 2Comment on Activity 2

Checkpoint 1Checkpoint 1

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