- Sophomore's dream
In mathematics, sophomore's dream is a name occasionally used for the identities
:discovered in 1697 by
Johann Bernoulli (especially the first).The name is in contrast to the "
freshman's dream " which is given to the mistake ("x" + "y")"n" = "x""n" + "y""n". (The correct result is given by thebinomial theorem .) Thesophomore 's dream has a similarly too-good-to-be-true feel, but is in fact true.Proof
We prove the second identity; the first is completely analogous.
The key ingredients of the proof are:
* Write "x""x" = exp("x" ln "x").
* Expand exp("x" ln "x") using the power series for exp.
* Integrate termwise.
* Integrate by parts.Expand "x""x" as
:
Thus by
termwise integration ,:
Evaluate the terms by
integration by parts ; integrate by taking "u" = (ln "x")"n" and "dv" = "x""m" "dx", which yields::
(also in the
list of integrals of logarithmic functions ).Thus inductively,
:
where ("n") "i" denotes the
falling factorial .In this case "m" = "n", and they are integers, so
:
Integrating from 0 to 1, all the terms vanish except the last term at 1(all the terms vanish at 0 because by
l'Hôpital's rule , and all but the last term vanish at 1since ln(1) = 0, which yields::
Summing these (and changing indexing so it starts at "n" = 1instead of "n" = 0) yields the formula.
References
*
Jonathan Borwein ,David H. Bailey , Roland Girgensohn "Experimentation in Mathematics: Computational Paths to Discovery" Page 44.
* William Dunham, "The Calculus Gallery, Masterpieces from Newton to Lebesgue", Princeton University Press, Princeton, NJ 2005, p. 46-51.
*N. J. A. Sloane ,OEIS|id=A083648 and OEIS|id=A073009
*Pólya andGábor Szegö , "Problems and Theorems in Analysis" (part I, problem 160).
* Weisstein, Eric W. [http://mathworld.wolfram.com/SophomoresDream.html "Sophomore's Dream."] From MathWorld--A Wolfram Web Resource.
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