Sophomore's Dream
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In mathematics, the sophomore's dream is the pair of identities (especially the first) \begin & \int_0^1 x^\,dx &&= \sum_^\infty n^ \\ & \int_0^1 x^x \,dx &&= \sum_^\infty (-1)^n^ = - \sum_^\infty (-n)^ \end discovered in 1697 by
Johann Bernoulli Johann Bernoulli (also known as Jean in French or John in English; – 1 January 1748) was a Swiss people, Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infin ...
. The numerical values of these constants are approximately 1.291285997... and 0.7834305107..., respectively. The name "sophomore's dream"It appears in . is in contrast to the name "
freshman's dream The freshman's dream is a name given to the erroneous equation (x+y)^n=x^n+y^n, where n is a real number (usually a positive integer greater than 1) and x,y are non-zero real numbers. Beginning students commonly make this error in computing the ...
" which is given to the incorrectIncorrect in general, but correct when one is working in a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
of prime characteristic with being a power of . The correct result in a general commutative context is given by the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
.
identity The
sophomore In the United States, a sophomore ( or ) is a person in the second year at an educational institution; usually at a secondary school or at the college and university level, but also in other forms of Post-secondary school, post-secondary educatio ...
's dream has a similar too-good-to-be-true feel, but is true.


Proof

The proofs of the two identities are completely analogous, so only the proof of the second is presented here. The key ingredients of the proof are: * to write x^x = \exp(x\ln x) (using the notation for the
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
and for the exponential function); * to expand \exp(x\ln x) using the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
for ; and * to integrate termwise, using
integration by substitution In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and c ...
. In details, can be expanded as x^x = \exp(x \log x) = \sum_^\infty \frac. Therefore, \int_0^1 x^x\,dx = \int_0^1 \sum_^\infty \frac \,dx. By
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E as the function domain i ...
of the power series, one may interchange summation and integration to yield \int_0^1 x^x\,dx = \sum_^\infty \int_0^1 \frac \,dx. To evaluate the above integrals, one may change the variable in the integral via the substitution x=\exp(-\frac). With this substitution, the bounds of integration are transformed to 0 < u < \infty, giving the identity \int_0^1 x^n(\log x)^n\,dx = (-1)^n (n+1)^ \int_0^\infty u^n e^\,du. By Euler's integral identity for the
Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
, one has \int_0^\infty u^n e^\,du=n!, so that \int_0^1 \frac\,dx = (-1)^n (n+1)^. Summing these (and changing indexing so it starts at instead of ) yields the formula.


Historical proof

The original proof, given in Bernoulli, and presented in modernized form in Dunham, differs from the one above in how the termwise integral \int_0^1 x^n(\log x)^n\,dx is computed, but is otherwise the same, omitting technical details to justify steps (such as termwise integration). Rather than integrating by substitution, yielding the Gamma function (which was not yet known), Bernoulli used
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
to iteratively compute these terms. The integration by parts proceeds as follows, varying the two exponents independently to obtain a recursion. An indefinite integral is computed initially, omitting the
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...
+ C both because this was done historically, and because it drops out when computing the definite integral. Integrating \int x^m (\log x)^n\,dx by substituting u = (\log x)^n and dv=x^m\,dx yields: \begin \int x^m (\log x)^n\,dx & = \frac - \frac\int x^ \frac\,dx \qquad\textm\neq -1\text\\ & = \frac(\log x)^n - \frac\int x^m (\log x)^\,dx \qquad\textm\neq -1\text \end (also in the
list of integrals of logarithmic functions The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. ''Note:'' ''x'' > 0 is assumed throughout this article, and the constant of integration ...
). This reduces the power on the logarithm in the integrand by 1 (from n to n-1) and thus one can compute the integral inductively, as \int x^m (\log x)^n\,dx = \frac \cdot \sum_^n (-1)^i \frac (\log x)^ where (n)_i denotes the
falling factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial \begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) . \end ...
; there is a finite sum because the induction stops at 0, since is an integer. In this case m=n, and they are integers, so \int x^n (\log x)^n\,dx = \frac \cdot \sum_^n (-1)^i \frac (\log x)^. Integrating from 0 to 1, all the terms vanish except the last term at 1,All the terms vanish at 0 because \lim_ x^m (\log x)^n = 0 by
l'Hôpital's rule L'Hôpital's rule (, ), also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives. Application (or repeated application) of the rule often converts an indeterminate form ...
(Bernoulli omitted this technicality), and all but the last term vanish at 1 since .
which yields: \int_0^1 \frac\,dx = \frac\frac (-1)^n \frac = (-1)^n (n+1)^. This is equivalent to computing Euler's integral identity \Gamma(n+1) = n! for the
Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
on a different domain (corresponding to changing variables by substitution), as Euler's identity itself can also be computed via an analogous integration by parts.


See also

*
Series (mathematics) In mathematics, a series is, roughly speaking, an addition of Infinity, infinitely many Addition#Terms, terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in ...


Notes


References


Formula

* * * *
OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...
, and * * * Max R. P. Grossmann (2017)
Sophomore's dream.
1,000,000 digits of the first constant


Function


Literature for x^x and Sophomore's Dream
Tetration Forum, 03/02/2010 *
The Coupled Exponential
'' Jay A. Fantini, Gilbert C. Kloepfer, 1998
Sophomore's Dream Function
Jean Jacquelin, 2010, 13 pp. * * * Footnotes {{reflist Integrals Mathematical constants