Content uploaded by Harlan J. Brothers

Author content

All content in this area was uploaded by Harlan J. Brothers

Content may be subject to copyright.

Novel Series-based Approximations to e

John A. Knox and Harlan J. Brothers

The College Mathematics Journal, September 1999, Volume 30, Number 4,

pp. 269–275.

John Knox (john.knox@valpo.edu) is an assistant professor of geography and meteorology at

Valparaiso University in Indiana. He received a B.S. in mathematics from the University of Alabama at

Birmingham and a Ph.D. in the atmospheric sciences from the University of Wisconsin-Madison. His

research interests revolve around the applications of mathematics to meteorology. He has also pub-

lished articles on the applications of storytelling concepts to the teaching of science courses. In his

spare time he composes music and an occasional poem, watches The Weather Channel with

climatologist-wife Pam, and plays golf with his two-year old son Evan.

Harlan J. Brothers (brotech@pcnet.com) is a professional inventor, design consultant, and amateur

mathematician based in Connecticut. He is the founder of BroTech. He currently holds five U.S.

patents and is developing proprietary applications for encryption technology. He is a jazz guitarist and

composer, having studied at the Berklee College of Music. He speaks Spanish and is also a black belt,

teaching the Japanese martial art of Aikido.

In this paper we dare to take one of the oldest dogs in the college calculus

curriculum—the Taylor series—and teach you how to do new tricks with it that

Newton, Euler and their successors do not seem to have discovered. Below, we use

this standard tool of introductory-level calculus to derive very accurate closed-form

approximations to e. The expressions we derive here and in a companion paper [2]

appear to be new, even though approximations to ewere first discovered in the 1600’s

[3, pp. 26-27]. Using our technique, you and your students can—with a little

perseverance—be able to derive and prove for yourselves entirely new and highly

accurate methods of calculating e.

An old stand-by of college calculus textbooks [1, p. 558; 4, p. 743] is the calculation of

evia

(1)

For example, inserting into (1) we get 2.71692 39322, which is eaccurate to

two decimal places.

Elsewhere in most college calculus textbooks [e.g., 1, p. 654; 4, p. 711] eis obtained

directly from the Maclaurin series for which is For

this equals

Although it is almost never done in calculus textbooks, it is a relatively simple task to

express the Classical approximation as a series. We can then evaluate the error of (1)

analytically rather than by trial-and-error.

11111

2! 11

3! 11

4! 1... 11

N!<e.

x51

e

x

511x1x

2

2! 1x

3

3! 1. . . .e

x

,

x51000

Classical:

1

111

x

2x

<e.

lim

x

→

`

s

111

x

d

x

:

First, express ln as a Maclaurin series, convergent for

[1, p. 635]:

Next, replace xwith and multiply the result by x:

Thus

or

(2)

The series in (2) shows the error in the Classical approximation. The approximation is

not very accurate for small xbecause it is only first-order accurate—that is, the series in

(2) possesses a term that is proportional to This term is relatively large for small x.

Since the series for the Classical approximation indicates where its Achilles’heel is in

terms of accuracy, we can use this information to create new algebraic expressions that

improve upon its accuracy.

The path to obtaining new and more accurate approximations to eusing series involves

repeated “bootstrapping,” i.e., combining two good approximations to get a better one.

We first obtain an approximation of eaccurate to second order, that is, of the

form

We can derive a second-order approximation to eby summing the two series for

5115

24x

2

25

24x

3

1... ;q

s

x

d

.

1

1

1

2x21

8x

2

11

24x

3

2...

2

x ln

1

111

x

2

1 ln

1

111

2x

2

5

1

121

2x11

3x

2

21

4x

3

1...

2

x ln

1

111

x

2

and ln

1

111

2x

2

:

e

3

11O

1

1

x

22

4

.

1yx.

x

≥

1.

1

111

x

2x

5e

3

121

2x111

24x

2

27

16x

3

12447

5760x

4

2959

2304x

5

1238 043

580 608x

6

2...

4

,

5e

3

11

s

p

s

x

d

21

d

1

s

p

s

x

d

21

d

2

2! 1

s

p

s

x

d

21

d

3

3! 1 ...

4

1

111

x

2x

5e

p

s

x

d

5e?e

p

s

x

d

21

5121

2x11

3x

2

21

4x

3

11

5x

4

21

6x

5

11

7x

6

2... ;p

s

x

d

, x

≥

1.

x ln

1

111

x

2

5ln

1

111

x

2x

1

x

ln

s

11x

d

5x2x

2

21x

3

32x

4

41x

5

52x

6

61x

7

72...

21

<

x

≤

1

s

11x

d

2

Thus

or

ACM:

(3)

We call this the “Accelerated Classical Method,” or ACM. Here and later, the acronym

refers to the closed-form expression on the left-hand side of the equation.

The series in (3) shows that ACM is more accurate than the Classical approximation,

because the power of xin the first term is larger; therefore the sum of the terms in the

brackets on the right-hand side of (3) is closer to 1 for all than in the case of (2).

However, every math classroom has its “Doubting Thomas” who must test the theory for

himself or herself. Fortunately, comparison of these approximations is easy with a

calculator. ACM is equal to the Classical approximation multiplied by

Therefore, for ACM is equal to 2.71692 39322 1.0005 2.71828 23942,

which is eaccurate to six decimal places. The calculator confirms the calculus; ACM is

superior to the Classical approximation!

(ACM was originally derived by numerically examining the relationship between the

Classical approximation and another new method. See Appendix A for this alternative

derivation, which includes additional material of use in introductory calculus courses.)

Another second-order-accurate approximation to eresults when we add the series for

to its “mirror image” with xreplaced by

The terms cancel in this summation, as do all the odd-powered terms in x. Dividing

the sum by 2 and exponentiating yields the “Mirror Image Method,” or MIM:

(4)

See [2] for more discussion of this method and its numerous extensions.

Now that we have two second-order-accurate methods for approximating e, we can

combine them to create a third-order-accurate approximation by summing the series

for ACM with the series for MIM, and then multiplying this sum by

The result is

(5)5e

3

125

9x

3

119

120x

4

277

180x

5

1137

1008x

6

2...

4

, x

>

1.

ACMMIM:

s

x11

d

11x

6

s

x21

d

5x

61

2x11

2x

x1128

3

8

3.25

8 3

MIM:

1

x11

x21

2

x

25e

3

111

3x

2

123

90x

4

11223

5670x

6

1...

4

, x

>

1.

1

2x

2x ln

1

121

x

2

5111

2x11

3x

2

11

4x

3

11

5x

4

11

6x

5

11

7x

6

1... .

2x:x ln

s

111

x

d

53x51000

s

111

2x

d

.

x

>

1

5e

3

115

24x

2

25

24x

3

11187

5760x

4

2587

2880x

5

1117 209

580 608x

6

2...

4

, x

≥

1.

1

111

x

2x1

111

2x

2

1

111

x

2x1

111

2x

2

5e?e

q

s

x

d

21

3

In Figure 1, we present a visual comparison of ACM, MIM, and ACMMIM with the

Classical approximation. [We emphasize for clarity that the left-hand sides of (3), (4),

and (5) are used in these calculations; the right-hand sides of these equations are simply

the analytical machinery that guided the creation of the closed-form expressions.] The

figure demonstrates that all three of these new methods significantly improve upon the

Classic closed-form expression.

Figure 1. A comparison of the new approximations ACM, MIM, and ACMMIM

versus the Classical approximation to efor

To summarize, our “bootstrapping” approach for deriving highly accurate closed-form

approximations to eis composed of three steps:

Step 1: Take the series for two (or more) algebraic expressions and add them together in

such a way that the lowest-power term in xin the sum of the series cancels out.

Step 2: Multiply both sides of this sum—the sum of the algebraic expressions on the

left-hand side, the sum of their series on the right-hand side—by a constant. This

coefficient (e.g., in ACMMIM) is dictated by what you need to multiply the right-

hand side with so that its constant term is exactly 1. Why do this step? Because in

Step 3, you will exponentiate both sides of the equation. If the constant term is 1, then

in Step 3 you will obtain and that’s what we’re after!

Step 3: Exponentiate both sides to get the final result. The left-hand side gives you a

closed-form algebraic expression that is a very accurate approximation to e, and the

right-hand side gives you a calculus-based quantification of how good this

approximation is.

We believe that many—one would hope most—college calculus students can grasp these

simple manipulations. Furthermore, approximations much more accurate than

ACMMIM can be obtained through this approach. For example, in Appendix B we cite

without derivation several new approximations to eobtainable via the approach outlined

e

11small terms in x

<e,

8y3

1

<

x

≤

10.

4

in this paper. In [2], we discuss how more exotic variants of MIM can lead to

exceptionally accurate approximations.

In conclusion, how often are you able to teach a 1990’s-vintage research result in

introductory calculus? We encourage teachers to present our results in the classroom

when you discuss the compound-interest formula, or when you introduce Taylor series.

Better still, our work provides an opportunity for you to challenge your students to think

about how their calculators and computers “know” constants such as e. By assigning

homework problems based on this paper, you can lead your students to discover for

themselves new approximations to esuch as ACM, MIM, those in Appendix B. . . or

perhaps some we haven’t discovered yet!

Acknowledgments. The authors thank Richard Askey, Lee Mohler, and David Ortland

for comments on this work, and Pam Naber Knox of National Public Radio’s “Science

Friday” staff for bringing the authors together in this collaboration.

References

1. H. Anton, Calculus, John Wiley and Sons, 1980.

2. H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic

constant e,Mathematical Intelligencer, Volume 20, number 4, 1998, pages 25-29.

3. E. Maor, e: The Story of a Number, Princeton University Press, 1994.

4. G. B. Thomas, Jr. and R. L. Finney, Calculus and Analytic Geometry, 7th edition,

Addison-Wesley, 1988.

APPENDIX A

Numerical Derivation of ACM

ACM was discovered by attempting to find a correlation between a new second-order

approximation to e, (B1) in Appendix B, and the Classical approximation (1). Letting

and Classical, the first step is to examine their ratio:

x

1 1.50000 00000

10 1.04845 38021

100 1.00498 34578

1000 1.00049 98335

[Note: since at is of the indeterminate form the evaluation of the ratio

at is an excellent opportunity to explore the utility of L’Hopital’s Rule by

comparing the analytic and numerical results.]

After subtracting 1 from this result for each generated value of xit becomes clear that

the remainder of this operation times xapproaches a value of 0.5 for large x:

x51

f

s

x

d

g

s

x

d

0

0,x51f

s

x

d

f

s

x

d

g

s

x

d

g

s

x

d

;f

s

x

d

;

s

B1

d

5

x

10.50000 00000

10 0.48453 80210

100 0.49834 57838

1000 0.49983 34583

This suggests that

(A1)

[A formal proof of (A1) is another good homework exercise!]

A new approximation with the same order of accuracy as (B1) can be obtained by

substituting for in this expression and solving

(A2)

for Doing so yields the closed-form expression for ACM.

APPENDIX B

Other Series-Based Approximations to e

The approximations below can be derived using the same bootstrapping approach

outlined in the text. Their derivations are left as exercises for the reader.

(second order) (B1)

(second order) (B2)

(third order) (B3)

These approximations can be combined with others found in the text to create the

following approximations to e:

(third order) (B4)

(fourth order) (B5)

(fourth order) (B6)

(fifth order) (B7)

656

75

s

B6

d

2581

75

s

B5

d

10

7

s

B3

d

23

7

s

B4

d

8

7

s

B1

d

21

7MIM

1

6ACM 15

6

s

B2

d

s

x11

d

x11

2x

x

1x

x

s

2x21

ds

x21

d

x21

2x

x

2

s

x21

d

x21

2x

x

s

2x21

ds

x21

d

x21

s

x11

d

1

111

x

2x

2

s

x21

d

1

121

x

22x

h

s

x

d

.

x

3

h

s

x

d

g

s

x

d

21

4

51

2

f

s

x

d

h

s

x

d

lim

x→

`

x

3

f

s

x

d

g

s

x

d

21

4

51

2.

x

3

f

s

x

d

g

s

x

d

21

4

6