Chapter 10
Polynomial and Series Representations of Functions





10.1 Sums and Limits

Exercises

  1. Calculate each of the following sums exactly (i.e., do not compute decimal approximations).
    a. `1+3+9+27+cdots+3^10` b. `1+1/3+1/9+1/27+cdots+(1/3)^10`
    c. `5+25+125+cdots+5^9` d. `3^6+3^7+3^8+cdots+3^12`
    e. `1+1/4+1/4^2+1/4^3+cdots+1/4^8` f. `1+2+4+8+16`
    g. `1+1/2+1/4+1/8+1/16` h. `1+1/2+1/4+1/8+1/16+cdots+1/1024`
    i. `1+1/2+1/4+1/8+1/16+cdots+1/2^12` j. `1/3^4+1/3^5+1/3^6+1/3^7+cdots+1/3^12`
    k. `7/13+(7/13)^2+(7/13)^3+cdots+(7/13)^20` l. `2/3+(2/3)^2+(2/3)^3+cdots+(2/3)^14`

  2. The NCAA Championship basketball tournament is a single elimination tournament that starts with 65 teams. The two lowest-ranked teams play each other in a "play-in" game, with the winner advancing to round one. The remaining 64 teams play in the first round, and the winners progress to the second round. This pattern is repeated until only one team is left undefeated.
    1. Express the number of games played (after the play-in game) as a geometric sum, and calculate this number.
    2. Describe a simpler method for determining the number of games played in the tournament.

  3. Suppose that 70 cents of every dollar spent in the United States is spent again in the United States. (Economists call this the multiplier effect.) If the federal government pumps an extra billion dollars into the economy, how much total spending in the United States occurs as a result?

    1. Use the geometric sum formula to explain why `0.3333...` is the same number as `1//3`.
    2. What fraction is the same number as `0.353535...`?
    3. What fraction is the same number as `0.200820082008...`?

    1. Find the exact sum: `1+6/7+(6/7)^2+(6/7)^3+cdots+(6/7)^40`.
    2. Find a decimal approximation to the sum in part (a) with 7SD accuracy.
    3. Find the exact sum with infinitely many terms: `1+6/7+(6/7)^2+(6/7)^3+cdots`.

  6. Suppose you drop a ball from a height of one meter, and the ball bounces to a height of `0.9` meter. (This is the right coefficient of restitution for a SuperBall.) You let the ball continue to bounce until it comes to rest.
    1. How high does it rise on the second bounce? On the third?
    2. Estimate the total distance that the ball travels up and down.
    3. Estimate the total time the ball is bouncing before it comes to rest.
    4. If you have a reasonably bouncy ball handy, you can check to see how close the theoretical model is to reality. First, determine what number should replace "`0.9`," and solve the exercise with a coefficient of restitution that matches your real ball. Compare the time result with your observation of total time.

  7. A golf ball has a coefficient of restitution of `0.78`. Suppose you drop a golf ball from a height of one meter.
    1. How high does it rise on the second bounce? On the third?
    2. Estimate the total distance that the ball travels up and down.
    3. Estimate the total time the ball is bouncing before it comes to rest.
    4. If you have a golf ball handy, check to see how close the theoretical model is to reality.

  8. Zeno of Elea, a Greek philosopher of the fifth century B.C., constructed several paradoxes to show the impossibility of motion. Here is one of them:
    You cannot walk across the room because, to do so, you would first have to walk half-way across the room, then half the remaining distance, half of that distance, and so on. As you have to cover half the remaining distance an infinite number of times, you will never complete the trip. Therefore, motion is an illusion.

    (If you are having any doubts about the wisdom of this argument, get up and walk across the room.)

    1. Using the width of the room as a unit of distance, how far do you walk in the first step of this process? In the first two? The first three?
    2. Without any further adding of fractions (but using your formula for calculating a geometric sum), how far do you walk in the first 27 steps?
    3. What happens when the number of steps becomes infinite? And just how is it that you can complete an infinite number of these steps in a finite time?

  9. In Chapter 1 we stated without explanation the following formula for the monthly payment on a new car:

    p = ( P - D )   r ⁢   ( 1 + r ) n ( 1 + r ) n - 1 ⁢ ,

    where

    p is the monthly payment,
    P is the price of the car,
    D is the down payment,
    r is the monthly interest rate (as a decimal fraction), and
    n is the number of months required to pay back the loan.

    Explain the formula. [Hints: `P-D` is the amount borrowed. The total amount paid back is `np`. The factor by which the outstanding balance grows each month (before a payment is made) is `1+r`.]

  10. Find each of the following limiting values. Start by exploring each of the functions with your graphing tool. Once you have a value, try to explain why it must be the right value. In some cases you may not have adequate tools — beyond the visual evidence — to establish that your answer is correct. We will consider some of these exercises again at the end of this chapter.
    a.   `lim_(x rarr 0) (tan 2x)/x` b.   `lim_(x rarr 0) (sin 2x)/(tan 3x)` c.   `lim_(x rarr 0) (tan^(-1) 5x)/(sin^(-1) 3x)`
    d.   `lim_(x rarr oo) (ln x)/x` e.   `lim_(x rarr 0) (sin^2 x)/x` f.   `lim_(x rarr 0) (cos x-1)/(x^2 e^x)`
    g.   `lim_(x rarr 0) (e^x - e^(-x))/(1-e^x)` h.   `lim_(k rarr oo) 5^k/(k!)` i.   `lim_(x rarr 0) (1/x-1/(e^x-1))`
    j.   `lim_(x rarr 0)  (text[(]sin x-xtext[)]^3)/(xtext[(]1-cos xtext[)]^4` k.   `lim_(x rarr oo) (tan^(-1) 5x)/(sin^(-1) 3x)` l.   `lim_(x rarr 0) 2^x/3^x`
    m.   `lim_(x rarr 0) (2^x-1)/(3^x-1)` n.   `lim_(x rarr oo) 2^x/3^x` o.   `lim_(x rarr 0) x ln|x|`
    p.   `lim_(x rarr pi//2) text[(]2x-pi text[)] tan x` q.   `lim_(k rarr oo) k^5/5^k` r.   `lim_(x rarr 0) (1/x-1/(sin x))`

  11. When we defined the definite integral in Chapter 7, we could have written the defining formula in the form

    ∫ a b f ( t )   d t = lim n → ∞   ∑ k = 1 n f ( t k - 1 )   Δ ⁢ t .

    Make sense of this terse notation by explaining carefully all the places that the index variable `n` is hidden in the expression on the right. Write out as explicitly as you can the first several terms of the sequence to which the "lim" notation is being applied.

  12. Sometimes we are interested in the behavior of a function as the independent variable approaches a number just from the right or just from the left. In particular, if the limiting values from left and right are different, then there is no two-sided limiting value. We use the notation `x rarr a^+` to indicate values of `x` approaching `a` from the right and the notation `x rarr a^-` to indicate values of `x` approaching `a` from the left. Determine each of the following limiting values. If you can, give a reason for your conclusion.
    a. `lim_(x rarr 0^+) |x|/x` b. `lim_(x rarr 0^-) |x|/x` c. `lim_(x rarr 0^+) x ln x`
    d. `lim_(x rarr 1^-) text[(]x-1text[)] ln text[(]x-1text[)]` e. `lim_(x rarr 0^+) text[(]x-1text[)] ln text[(]x-1text[)]` f. `lim_(x rarr 1^-) (ln x)/(1-x)`
    g. `lim_(x rarr -1^+) text[(]x-1text[)] ln text[(]x-1text[)]` h. `lim_(x rarr 0^+) (1/x-1/sqrt(x))` i. `lim_(x rarr 1^+) (ln x)/(1-x)`

  13. We can use the notations `oo` and `-oo` to describe behavior of function values as well as values of the independent variable. For example, we write

    `lim_(x rarr 0^+) 1/x=oo`   and   `lim_(x rarr 0^-) 1/x=-oo`

    to describe the limiting behavior of the reciprocal function as `x` approaches `0` from the right and from the left, respectively. Describe the limiting behavior represented by each of the following notations. If there is a limiting value, state the value. Whether there is a limiting value or not, give a reason for your conclusion.
    a.   `lim_(x rarr 0) 3/|x|` b.   `lim_(x rarr 0) 2/x^2` c.   `lim_(k rarr oo) 5^k/k^5`
    d.   `lim_(x rarr oo) e^x/x` e.   `lim_(x rarr -oo) e^x/x` f.   `lim_(x rarr 0^-) (sin x)/x^2`
    g.   `lim_(x rarr 0^+) (ln x)/x` h.   `lim_(x rarr 1^-) (ln text[(]1-xtext[)])/(1-x)` i.   `lim_(x rarr oo) (1/x-1/sqrt(x))`
    j.   `lim_(x rarr (pi/2)^-) (x-pi) tan x` k.   `lim_(x rarr (pi/2)^+) tan x` l.   `lim_(x rarr 0^-) (1/x-1/e^x)`

  14. Find each of the following limiting values. You may use a computer or calculator.
    a.   `lim_(x rarr oo) x^(-1//x)` b.   `lim_(x rarr oo) (sin x)/x` c.   `lim_(x rarr 0) text[(]1+xtext[)]^(1//x)`
    d.   `lim_(x rarr oo) (ln x)/x^2` e.   `lim_(x rarr oo) x^2/x^x` f.   `lim_(x rarr 0^+) x^2/x^x`
    g.   `lim_(x rarr oo) (sqrt(1+x^2)+1)/x` h.   `lim_(x rarr 0^-) (sqrt(1+x^2)+1)/x` i.   `lim_(x rarr 0) text[(]1-xtext[)]^(1//x)`

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