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The Convergence Behavior of fα (x ) = (1 + x1 )x +α
Cong X. Kang ([email protected]) and Eunjeong Yi ([email protected]), Texas A&M
University at Galveston, Galveston, TX 77553
Duane W. DeTemple provided the proof of the monotonicity of (1 + n1 )n and
(1 + n1 )n+1 using integration [2]. His article follows the appearance of a number of
earlier papers on sequence convergence to e using a variety of methods: Bernoulli’s
Inequality [1], the Arithmetic-Geometric Mean Inequality [3], Hadamard’s Inequality
[4], the Binomial Theorem [5], and the Mean Value Theorem [6].
Set f α (x) = (1 + x1 )x+α . Notice
1 x
1 α
lim f α (x) = lim 1 +
· lim 1 +
=e
x→∞
x→∞
x→∞
x
x
for any α ∈ R. We classify the convergence behavior of the one-parameter family
f α (x) = (1 + x1 )x+α , thus obtaining an elementary example that’s in keeping with the
general principle—especially espoused by Grothendieck in algebraic geometry—that
an object only reveals itself fully in “dynamics”. Notably however, we use nothing
more than what is taught in introductory calculus courses. Our main result is the following theorem.
Theorem. Let α ∈ R, and let f α (x) = (1 + x1 )x+α for x > 0.
(a) If α ≤ 0, then f α is strictly increasing.
(b) If α ≥ 12 , then f α is strictly decreasing.
(c) If 0 < α < 12 , then f α has exactly one local minimum and it is the absolute
minimum.
Proof. We make repeated use of the following observation: If g is continuous and
strictly monotonic on (l, r )—where either endpoint may be infinite, if g(x0 ) > 0 for
some x0 ∈ (l, r ), and if g(x) approaches 0 as x approaches either endpoint, then
g(x) > 0 for all x ∈ (l, r ). (Of course, a similar statement holds if both inequalities
are reversed.)
Now, by differentiating and using a little algebra, we find that
1 x+α
1
x +α
f α (x) = 1 +
ln 1 +
− 2
.
x
x
x +x
Since the first factor is always positive for x > 0, we only need to consider the behavior
of
1
x +α
.
(1)
− 2
gα (x) = ln 1 +
x
x +x
Notice lim gα (x) = 0, a fact to be used in all three cases which follow.
x→∞
By differentiating (1), we have the following:
gα (x) =
(2α − 1)x + α
.
(x 2 + x)2
(2)
α
The only critical number is x0 = 1−2α
, which lies within the domain (0, ∞) only when
1
α ∈ (0, 2 ). We consider three cases.
VOL. 38, NO. 5, NOVEMBER 2007 THE COLLEGE MATHEMATICS JOURNAL
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Case 1: α ≤ 0. It is easy to check that gα (1) > 0 and gα is strictly decreasing. Therefore, by our observation, gα (x) > 0 for all x > 0. Hence, f α (x) > 0 for x > 0,
which implies f α (x) is strictly increasing. Further, lim f α (x) = 1 for α = 0, and
x→0+
lim f α (x) = 0 for α < 0 (see Figure 1).
x→0+
y=e
y = f α (x)
y=e
y = f α (x)
α=0
α<0
Figure 1. Graph of f α when α ≤ 0.
Case 2: α ≥ 12 . It is easy to check that gα (1) < 0 and gα is strictly increasing. Therefore, by our observation, gα (x) < 0 for all x > 0. Hence f α (x) < 0 for x > 0, which
implies f α (x) is strictly decreasing. Further, lim f α (x) = ∞ (see Figure 2).
x→0+
y = f α (x)
y=e
Figure 2. Graph of f α when α ≥ 12 .
Case 3: 0 < α < 12 . From (2), it is easy to check that gα has a local maximum at
α
its only critical number x0 = 1−2α
. Looking at gα (x0 ) = ln( 1−α
) − 2(1 − 2α) as a
α
function h of α, it is easily verified that h( 13 ) > 0, h (α) < 0, and lim h(α) = 0.
α→ 12
−
We apply our observation to h to conclude that gα (x0 ) > 0 for 0 < α < 12 . We apply
the observation again on the restricted domain (x0 , ∞) to conclude that gα (x) > 0 for
x ≥ x0 . Additionally, lim gα (x) = −∞. To see this, we use l’Hospital’s rule:
x→0+
1
lim e gα (x) = lim
x→0+
386
x→0+
e
x+α
x 2 +x
·
x 2 +2αx+α
(x+1)2
= 0, since α > 0.
c THE MATHEMATICAL ASSOCIATION OF AMERICA
y=e
y = f α (x)
β
Figure 3. Graph of f α when 0 < α <
1
2
The preceding tells us that gα has exactly one zero, say β, between 0 and x0 , that
gα is strictly increasing on (0, x0 ), and that gα is strictly decreasing and positive for
x ∈ [x0 , ∞). To summarize:
• If 0 < x < β, then g (x) < 0 and f is strictly decreasing.
α
α
• If β < x < ∞, then g (x) > 0 and f is strictly increasing.
α
α
1
Therefore, if 0 < α < 2 , then f α has exactly one local minimum—in fact, the absolute
minimum (see Figure 3).
Acknowledgment. The authors greatly appreciate the comments and corrections from the editorial board and the
referees.
References
1. R. B. Darst, Simple proofs of two estimates for e, Amer. Math. Monthly 80 (1973) 194.
2. D. W. DeTemple, An Elementary Proof of the Monotonicity of (1 + n1 )n and (1 + n1 )n+1 , College Math. J. 36
(2005) 147–149.
3. N. S. Mendelsohn, An application of a famous inequality, Amer. Math. Monthly 58 (1951) 563.
4. P. R. Mercer, On the monotonicity of (1 + n1 )n and (1 + n1 )n+1 , College Math. J. 34 (2003) 236–238.
5. J. F. Randolph, Basic Real and Abstract Analysis, Academic Press, 1968.
6. N. Schaumberger, Another application of the mean value theorem, Two-Year College Math. J. 10 (1979) 114–
115.
From the section on Compound Addition in The American Tutor’s Assistant Revised; or,
A Compendious System of Practical Arithmetic, Joseph Crukshank, Philadelphia, 1809,
p. 30.
M OTION OR C IRCLE M EASURE .
This is used by astronomers, navigators, &c.
The denominations are:
60 seconds ( ) make . . . . . . . . . . . . . . . . . 1 minute 60 minutes. . . . . . . . . . . . . . . . . . . . . . . . . .1 degree ◦
30 degrees . . . . . . . . . . . . . . . . . . . . . . . . . . 1 sign sig.
12 signs, or 360 degrees, one revolution, or circle.
(See also page 377.)
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