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Novel Series-Based Approximations to e

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John A. Knox at University of Georgia
John A. Knox
Harlan J. Brothers
Harlan J. Brothers
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In this paper we take one of the oldest dogs in the college calculus curriculum - Taylor series - and teach you how to do new tricks with it that Newton, Euler and their successors do not seem to have discovered.
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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
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