Sophomore's dream

In mathematics, sophomore's dream is a name occasionally used for the identities (especially the first)

$\begin{align} \int_0^1 x^{-x}\,dx &= \sum_{n=1}^\infty n^{-n}&&(\scriptstyle{= 1.29128599706266354040728259059560054149861936843\dots)} \\ \int_0^1 x^x \,dx &= \sum_{n=1}^\infty (-1)^{n+1}n^{-n} = - \sum_{n=1}^\infty (-n)^{-n} &&(\scriptstyle{= 0.78343051071213440705926438652697546940768199014\dots}) \end{align}$

discovered in 1697 by Johann Bernoulli.

The name "sophomore's dream", which appears in (Borwein & Bailey 2004), is in contrast to the name "freshman's dream" which is given to the incorrect equation (x + y)ⁿ = xⁿ + yⁿ. The sophomore's dream has a similar too-good-to-be-true feel, but is in fact true.

Proof

Graph of the functions y = x^x and y = x^−x on the interval x ∈ (0, 1].

We prove the second identity; the first is completely analogous.

The key ingredients of the proof are:

• Write x^x = exp(x log x).
• Expand exp(x log x) using the power series for exp.
• Integrate termwise.
• Integrate by substitution.

Expand x^x as

$x^x = \exp(x \log x) = \sum_{n=0}^\infty \frac{x^n(\log x)^n}{n!}.$

Thus by termwise integration,

$\int_0^1 x^xdx = \sum_{n=0}^\infty \int_0^1 \frac{x^n(\log x)^n}{n!} \, dx.$

To evaluate the above integrals we perform the change of variable in the integral $\scriptstyle x=\exp\, {-\frac{u}{n+1}}$, with $\scriptstyle 0 < u < \infty$, so the integral

$\int_0^1 x^n(\log\, x)^n dx$

writes

$(-1)^n (n+1)^{-(n+1)} \int_0^\infty u^n e^{-u} du\, .$

By the well-known Euler's integral identity for the Gamma function

$\int_0^\infty u^n e^{-u} du=n!$

so that:

$\int_0^1 \frac{x^n (\log x)^n}{n!}\; dx = (-1)^n (n+1)^{-(n+1)}.$

Summing these (and changing indexing so it starts at n = 1 instead of n = 0) yields the formula.

References

Formula

• Jonathan Borwein, David H. Bailey, Roland Girgensohn Experimentation in Mathematics: Computational Paths to Discovery 2004, Page 44.
• William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton, NJ 2005, p. 46-51.
• OEIS, (sequence A083648 in OEIS) and (sequence A073009 in OEIS)
• Pólya and Gábor Szegő, Problems and Theorems in Analysis (part I, problem 160; 1998, p. 36)
• Weisstein, Eric W. Sophomore's Dream. From MathWorld—A Wolfram Web Resource.

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, 1998, 13 pp.
• Lehmer, D. H. (1985). "Numbers associated with Stirling numbers and x^x". Rocky Mountain Journal of Mathematics 15: 461. doi:10.1216/RMJ-1985-15-2-461. 
• Gould, H. W. (1996). "A Set of Polynomials Associated with the Higher Derivatives of y = x^x". Rocky Mountain Journal of Mathematics 26: 615. doi:10.1216/rmjm/1181072076.