Infinite Exponentials
Author(s): D. F. Barrow
Source: The American Mathematical Monthly, Vol. 43, No. 3 (Mar., 1936), pp. 150-160
Published by: Mathematical Association of America
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150
[March,
INFINITE EXPONENTIALS
INFINITE
EXPONENTIALS
By D. F. BARROW,University
of Georgia
1. A continuedexponentialshall be denoted by the symbol
Es-oai
whichis an abbreviationfor
.an
a,
aO
a2
that is, each ai is used as the exponentof the preceding.This symbolis selected
because of its analogy with the symbols fora continued sum and a continued
product,namely
n
E
i-O
n
ai
and
H ai.
i=O
In the continued exponential, the various ai shall be called exponents.A
longersymbol which puts the exponents into evidence will sometimesbe used
when one or moreof the exponentsare irregular,namely
E(ao, a,, a2,
. . .,
an)-EE,-oai.
The orderof thisexponentiallisn.
2. Some obviouspropertiesof continuedexponentialsare
(a)
E(ao, al, ...
,
am, am+,, ...
(b)
E(ao, ai,
,
am, 1, am+2,
(c)
E(ao, a,. * *
, am,
0,
am+2,
.
,
an) = E[ao, a,, * * -, am, E(am+i,
, am)
aan) = E(ao, al, .
, a,n) =
E(ao, al, *
, an)]
, am-1)
3. An infiniteexponentialis naturallydenoted by the symbol
E??=Oaz or
E(ao, al,,
We shall study only the case where all theai are positiveor zero,so that none
of the exponentialsconsideredlater is negativeor complex.
If the limit,as n-*o, of Ei=Oaiexists and equals say k, we shall say that the
infiniteexponentialE,l0ai convergesand has thevalue k; but if the above limit
does not exist,the infiniteexponentialshall be said to diverge.
The infiniteexponential Ej_aj shall be called the mth residual of E,0Oaj.
If every residual.of an infiniteexponential converges,then the infiniteexponential shall be called properlyconvergent;
and if an infiniteexponentialcon-
19361
INFINITE EXPONENTIALS
151
verges but at least one of its residuals diverges,the infiniteexponential shall
be called improperly
convergent.
4. Some obvioustheoremsabout infiniteexponentialsare:
THEOREM
1. If any one oftheresidualsconverges,
theinfiniteexponentialcon-
THEOREM
2. If any particular residual diverges,then everylater residual
verges.
diverges.
If E70ai grows infinitewith n, the correspondinginfiniteexponential shall
be called properlydivergent;but ifE,0Oaineverexceeds a fixedbound as n grows
infinite,and yet fails to approach a limit,then the infiniteexponentialshall be
called improperly
divergent,
or oscillating.
THEOREM 3. If some exponentin an infiniteexponentialis less than unity,
theexponentialmust convergeor oscillate. For if am is less than unity, the mth
residualwould not be greaterthan unity,and so could not growinfinite.
THEOREM 4. If all the exponentsin an infiniteexponentialare greaterthan
unity,thevalue ofE_.oai is monotoneincreasingwithn.
These two theoremsshow that the two typesof divergencedefinedabove are
all inclusive,that is, thereis no type of unbounded oscillation.
5. The nth margerinfunctionof x shall be definedby E(ao, a,, ... , an, x)
and its propertiesare developed here to be used in provingconvergencetheoremsforinfiniteexponentials.
6. A graphical process for evaluating the margerinfunction(or any continuedexponential) may be describedas follows:
Plot the graphsof the curves
yad,, y =-all, ..** y = ax$
(They are indicated in fig. 1 by the curves Co, Cl, C2, C3, C4, and the other
curves are to be disregardedforthe moment). Draw also the forty-five
degree
turnerline.
Then choose a point P upon the turnerwhose abscissa is x, fromP draw a
vertical line to meet C,,in the point Q,,,fromQndraw a horizontalline to meet
the turnerin P,, fromPn draw a vertical line to meet Cn-1 in Qn-1,fromQni1
draw a horizontalline to meet the turnerin P,,-, and continue until we have
used each of the curves in turn beginningwith Cn and endingwith CO,and the
finalpoint Po, on the turner,will have an abscissa equal to the desiredvalue of
the nthmargerinfunction.
If we startwith an infinitesimal
segmentPP' of the turnerwhose horizontal
projectionis Ax, and carry its points throughthe above graphical process,we
shall end with anotherinfinitesimal
segmentPoP', whose horizontalprojection
we shall call Axo; and moreoverat each step of the process, the lengthof the
152
[March,
INFINITE EXPONENTIALS
segment is multipliedby the slope of the C-curve taken at the corresponding
Q-point.Then if we let mi denote the slope&ofCi at the point Qi, we shall have
AxO = mO*ml
...
mn-AX
and hence the slope of the margerinfunctionis given by
d
-
dx
E(ao, a,
...
,
an),x)
mOm-l*... Mn.
/
y
of y=ax
Graphs
for
6
-
gee
aze
eo.0.I0.S50,O.7',oq9,I.I,
1.3 a.P
e Ye
/
21
/
/
Also x-=?y
logey
1
~~~2
e. X
FIG. 1
The foregoing
discussionmakesit desirableto markoffthe regionof the
plane in whichthe curvesof the familyy=ax have slopesnumerically
greater
thanunity,and theregioninwhichtheyhaveslopesnumerically
lessthanunity.
We shall call the formerregionthe expandingregion,and the latterthe contracting
region.These two regionsare separatedby a curvecalled the neutral
locus,at each pointofwhichsomememberofthefamilyhas a slopeofplusone
or minusone.
To gettheneutrallocuswe set
1936]
153
INFINITE EXPONENTIALS
dy
_
dx
=
ax
log a
=
?1
fromwhich the equation of the neutrallocus is foundto be
x = + y log y.
Fig. 1 shows part of the familyand the neutral locus, and it is easy to see the
expandingand contractingregions.
7. The problemof the convergence
of infiniteexponentialsall of whose exponentsare equal is now easily solved.* Consider E(a, a,
). If one approach
the question of the convergenceof thisinfiniteexponentialby evaluating graphically the series
E(a), E(a, a), E(a, a, a),
he will use only the graph of y=ax and the turner(see figs.2, 3, 4, 5), and will
be led to announce the followingtheorem:
THEOREM 5. (a) If elle <a, thenE(a, a, a,
) is properlydivergent.(b)
If e-e?a ?ele thenE(a, a,
and has thevalue k whichsatis) is convergent
) is improperly
then E(a, a,
fies k=ak and 1/e<k?e. (c) If O<a<ee,
divergentand as the numberof exponentsincreases in E(a, a,
, a) this exponential tendsto oscillatebetweentwo values ki and k2 neverassuming a value
betweenthem,wherek, and k2are determined
bysatisfying
ki =
ak2
and k2 =
ak,
and
O < k, <-
1
e
< k2 < 1.
Figures 2, 3, and 4 illustrateparts (a) and (b) of the above theorem,and are
selfexplanatory.Fig. 5 illustratespart (c). In this figure,the pointsQ., Qn-2, *
approach from the left a limitingposition Q, and Qnl, Qn-3 .* * approach
fromthe righta limitingpositionQ'. The pointsQ and Q' are located as the only
points on the graph of y =ax which are symmetricalwith respectto the turner;
and two such points will exist only when a <e-e. Theorem 5 bringsinto prominence two numbersoffundamentalimportancein our problem,namely
el/e =
1.444668...
e-e =
.06598803
which we shall call theupper and lowerlimitsof theintervalof convergence.
The
exponentialcurve y = exieis tangent to the turnerat the point (e, e) and y = e-ex
cuts the turnerat rightangles at the point (1/e, 1/e). The membersof the exponential familyin between these two cut the turnerat slopes numericallyless
* Sincecompleting
thepaper,it has cometo theattentionoftheauthorthatProf.Edmund
Landau has treatedthecase ofequal exponents
inhisuniversity
lectures,
and obtainedtheresults
givenin theorem5; but thereappearsto be nothingpublisheduponthesubject.
154
[March,
INFINITE EXPONENTIALS
5y
Y
2
Graphof y=(i.~
~ I14
~
~~~Grp)?f
~~
P is fixed RPrecedes
as n. incroases, since
/
I
/XP
&
4 a-s r. inueases.
g
sirLeC
e-e 1.4< e3'
(e, 1.5
0
~
~
______________________
7C
~
~
~
3
,
~~~~
-7
_____________________
FIG. 2
2
X
FIG. 3
Graph of yx(.o5)$
P is fixed.P0tenclsto
oscillate,as x increases
Since
05 <e-e/
_Grapkofyz(.sY)P Is fixed. Po
a.pproaches
Aas,
lncreases,sincer
eF'.
t
Q
4 FGe.
ill/
I
IiQ
5
__p t
FIG. 4
FIG. 5
than unity,and this is the reason why the exponential in theorem5 (b) converges.
19361
155
INFINITE EXPONENTIALS
8. Lemmas concerning
themargerin
function.
LEMMA 1. The margerinfunctionE(ao, al,
, an, x) is a monotoneincreasingor decreasingfunctionof x accordingas an evenor an odd numberof the
a, are less thanunity.For
d
-E(ao,
dx
=
al, ..
E(ao, al, ...
,nx)
,
an, x) E(a1, a2,
*a,
X)
...
E(an, x) log ao log a,...
log an
and all factors in this product are positive except the logarithmsof those ai
whichare less than unity.
LEMMA
2. If all the ai lie in the interval of convergence, that is if
e-e <_ a, :_ e1/e
i = 0, 1, 2,
and if 0?< x < e, then
O < E(ao, al,***,
an) x)
< e.
The graphical process, fig. 1, will make this self evident.
LEMMA
3. If all the ai are in the interval of convergence, that is if
e-e < ai < eIe,
i = 0, 1, 2,** *
then the difference between the largest and smallest values of E(ao,
in the interval 0 < x < e approaches zero as n grows infinite.
a,,
, an,
x)
Since the margerinfunctionis monotone increasingor decreasing,we can
get the differencebetween its largest and smallest values by integratingits
derivative over the interval. Using the formof this derivative given in Art 6,
we are tryingto prove
lim
MO-Ml *... Mn*dx = 0.
Now referringto fig. 1, and rememberingthat each mi is the slope ofa
member of the familyat the point Qi, we note the followingfacts. If a point
such as Q, or Q4 fall in the contractingregion,the correspondingslope ml, or
M4 is numericallyless than unity. But if a point like Q2 fall in the expanding
region,the value of M2, is numericallygreater than unity. However the preceding point Q3 must fall in the contractingregion,as is easily seen, and we
shall show that the productm2m3
is less than unity.Let R2 and R3 be the points
where the lines P3Q2 and P3Q3 meet the curve y = e-ex, and we easily see that
the slopes of this curve at R2 and R3 have greater numerical values than m2
and M3 respectively;and furthermore,
by a littleanalytic geometry,we see that
the productof the slopes of thiscurve at R2 and R3 is less than unity.This prod-
156
[March,
INFINITE EXPONENTIALS
uct may, howeverapproach unity as P3 approaches the point where the turner
cuts the last mentionedexponentialcurve.
Thus the integrand consists (with the possible exception of a finitelast
factor) of factorsand pairs of factorseach of which is less than unity. We wish
to show that, as n grows infinite,the integrandapproaches zero at all points
of the intervalwith the possible exceptionof one point where it cannot exceed
unity. The least favorable case occurs when all the ai are equal to e-e and x is
near the value 1/e,or else when all the ai are equal to elIe, and x is near the value
e. For at these points the extremecurves in fig.1 are meeting the turnerwith
slopes of - 1 and 1 respectively.But even in these cases a small segmentwill
be indefinitelyshortenedby the graphical process since each of its end points
must approach the point where the exponential curve meets the turner.Since
the integrandmay be made arbitrarilysmall by increasingn, except perhaps
at one point, the value of the integralmay be made arbitrarilysmall, and our
lemma is proved.
0
yz (e-ee%
I
y(Ce)(e)1
-
iee)y(ee)
2
0y=(eJ
e
FIG.6
(ei~~~~~~~~~~~~~~~~~(
FIG.7
To clarifythis discussion,Fig. 6 shows the graphs of the firstfew margerin
functionswhenall of the at are equal to the lowerlimite-e,and Fig. 7 shows them
when all the as are equal to the upper limitelle. One can see how the difference
between the largest and smallest values of the margerinfunctionapproaches
zero, although the slope at one point is 1 or -1.
9. Fundamentaltheorems
on theconvergence
ofinfiniteexponentials.
THEOREM 6. The infiniteexponentialE= 0a1converges
if m can be found such
that
1936]
157
INFINITE EXPONENTIALS
e-e?_ ai < elle
i.=m,m+1,m+2,
.
Stated in words the theoremsays that if beyond a certain point all the exponents lie in the interval of convergence,then the infiniteexponential converges. By theorem 1 it will be sufficientto prove the convergenceof the mth
residual E=m ai. By lemma 3, we can choose n so large that the total variation
of E(am, am+1, *
, a
x) as x ranges from0 to e shall be less than e, an
arbitrarilysmall positive number.Then by lemma 2 the value of the finiteexponentialE"i+ ai will be altered by less than e ifwe make n larger.This proves
the theorem.
Theorem 6 is somewhat analogous to the fact that an infiniteproduct of
positive factorsconvergeswhen the factorsare all less than or equal to unity.
And the furtherfact that the infiniteproduct convergeseven when the factors
all exceed unityby small amounts which forma convergentseries suggeststhat
we investigatewhetherthe infiniteexponentialmay not convergewhen all the
exponents exceed the upper limit elle or are less than the lower limit e-e by
small amounts. The results were surprisingin that the convergenceof the
"small amounts" as a series was not sufficientfor the convergenceof the infiniteexponential.
7. The infiniteexponentialEtXO(elIe+ej),wherethe sE are all posior
tive zero,(a) will converge
if
THEOREM
elle
lim
enn2
n--+ao
<
lim enr2
>
(b) and will divergeif
2e
el/e
-.
2e
4. Any properlyconvergent
infiniteexponentialcan be put in theform
. . ), whichwe shall call the b-form.For we need only choose
each bi equal to the ith residual.
LEMMA
E(bolIbl, blll2, .
LEMMA 5. Any exponentialin the b-formconverges
if each bi is greaterthan
unity.To prove this considerthe finiteexponentialE(bollbl, bl11b2, . I bnlbn+).
It steadily increases with n since each exponentexceeds unity. Furthermoreit
neverexceeds the value bo,forifwe increase the last exponentby replacingbn+l
by 1, the whole symbolevidentlytelescopesdown to the value bo.
Thereforeto prove theorem7, it is necessary and sufficientthat it can be
put in the b-formwitheach bi greaterthan unity,that is, that
elle + en
b I/bn+l
Now since the maximumvalue of xllxis elle, in orderthat en be positive forall
values of n it is necessarythat bn> bn+l> e. We accordinglywrite:
elle +
En =
(e
+ bjn)l(e+8n+1)
158
INFINITE
[March,
EXPONENTIALS
Expanding by Taylor's serieswe have
n=
f
1
ele
-
(
+2
(bn
2e4}
1
-
5n)
an+1)2
[25n+l(bn
-n+l)
+
52 1
+***}
The principalpart of enis containedin
eI
-
n
(n-
Le2
n.
an+l)
2e3
To make the principalpart of e,npositiveand ofas low orderas possiblewe must
have the principalpart of Anequal to k/nwhere k is some constant. So we give
Anthis value, and findthat the principalpart of enhas a maximumof el/1/(2en2)
when k equals e.
Now to prove part (a) of theorem 7 we say that an exponential in the bformcan be foundas above, and its exponentswill be largerthan el/"+Enso the
given exponential converges by comparison. But in part (b) of theorem 7 no
exponentialin the b-formcan be foundwhose exponentsare as large as elle+ n,
so thisone cannot be convergent.
8. That theinfiniteexponentialE, o(e-e- ei) shall converge
properly,
whereei are all positiveor zero,and i ! ei+, it is necessarythat1imj_e =0 and
it is sufficient
thatliM,_xiqC_= 0 whereq > 1.
THEOREM
A proofof this theoremmay be carriedthroughby a carefulanalysis of the
graphicalprocess,and is omittedbecause it is lengthyand tedious.
10. Whentheexponentsof an infiniteexponentialare functionsof x, the exponential will define a functionof x for all values of x which render it convergent.We shall study one such case, namely
f(x) = E(e kox, eklx .2 .
.
When x is zero this functionis equal to unity. Assuming that there are other
values forwhich the exponential converges,it can be developed formallyinto
the followingseries:
E(kox
ekiz
..
x2
I + kox+ (kJ2+ 2koki)2
+ (ko3+ 6kd2ki+ 3kok2+ 6kokik2)-3
3!1
+ (k04 + 12ko ki + 24k Ikj? + 4kokl' + 24kek1k2
+ 24kok?k2+ 12kokik22
+ 24kok1k2k3)
1936]
INFINITE
159
EXPONENTIALS
+ 60ko'k1k2
+ (ko'+ 20k04k1
+ 90ko3
ki?+ 80k2 ki3+ 5kok14
+ 240k2 k?k2+ 60k12
k1k2?
+ 60kokis
k2+ 120kok2ke?
2
120k
120kok
+ 20kok1k2?
+
2k2k3+ 120koklk1
+
k1k2k3
k,
+ 60kok1k2k1? +
x5
120kok1k2k3k4) - +
The generalexpressionfora termin any one of theseparenthesesis
(ao + oi +
+ a,)!
a,
a2
atO 'al
..
ao!a
2! ..
ap
Z
a0 al
aP.
koki
a2
k2
...
ap
kp
If all the k, are equal to unitywe obtain
E (e,
,
)=1
+ x+
3
-
2!
X2+
16
- XI +
(n + )W-
,+
3!
n +**e
and the ratio test shows that this series convergeswhen -1/e<x < 1/e. But
theorem6 shows that the exponentialconvergeswhen - e < x ? 1/e.
This functionsatisfiesthe transcendentalequation
log y = xy
and may be regardedas a means of solvingthisequation explicitlyfory in terms
of x. Since the ratio of each termto the precedingis always less than ex, it follows that the remainderof the series beyond the nth termis less that the geometricseries
x (1+
ex+enx+*
n!
n)
-
+)-
xn(I + eX + e2X2+
n!
1-ex
11. A n arbitraryfunctionof x whosevalue is unitywhenx is zero can be developedinto an infiniteexponentialofthetypeconsideredin article10 by equating
the termsin its series developmentto the termsin the developmentof the exponential and determiningthe values of the ki one afteranother. For instance:
cos x
= E(e_z212,ez216,e11Z2160 . . . )
sin x
= xE(e-x216, ex2130,e59x2/1260. . .)
The author was keenly disappointed in not findingeither simple values for
the k, or a general expression for the nth exponent, especially since these
functionshave such elegantexpansionsas infiniteseriesand infiniteproducts.
12. The solutionfory in termsofx of the transcendentalequation*
* H. L. Slobin,thisMONTHLY, Oct. 1931,p. 444,solvedthisequationintermsofa parameter.
He was interested
infinding
positive,rational,unequalvaluesofx and ythatsatisfytheequation,
oneobviouspairbeing2 and 4.
160
[March,
ISOGONAL AND ISOTOMIC CONJUGATES
yZ
=
xv
may be carriedout by firstwritingit in the form
y =
(xl/x)y
and thensubstitutingthe whole rightside of this equation forthe y that stands
in the exponent,getting
(XlI/X) (x
y -
):Y.
Continuedsubstitutiongives the infiniteexponential
y = E(xlIx,
XllX, . . .
)
whichconvergesforall values ofx greaterthan 1/e.
Y
y,
-
4~~~~~~~~~~1e
5
e)
//
1 4 ,t
of y"=xyis entirefigure
~~Graph
Gtaph of y
O , (W,+)
1
2
is solidlineonJ1.
4
3
5
6
7 X
FIG. 8
The graph of thisinfiniteexponentialhas a ninetydegreecornerat the point
(e, e). The solid line in Fig. 8 is the graph of the infiniteexponentialwhilethe
whole curve is the graph of the originalequation forpositive values of x and y.
ISOGONAL AND ISOTOMIC CONJUGATES AND THEIR
PROJECTIVE GENERALIZATION
By P. H. DAUS, Universityof Californiaat Los Angeles
1. Introduction.Let P be a point in the plane of the triangleA1A2A3,and
designate the sides of the triangleby ai and the lines A iP by pi. If p/ is determined so that angle a3Pl-angle p la2, sense taken into account, then pi and
P' are called isogonal conjugate lines. The followingfundamental incidence
propertiesfollowimmediatelyfromthe trigonometricformof the theoremsof
Ceva and Menelaus.