Sophomore's Dream

Author

Eric W. Weisstein
June 27, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/SophomoresDream.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SophomoresDream.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

(∫_0^1)(x^x)dx

Integral

Integrate[(x^x),x]

Integrate[(x^x),{x,0,1}]

Integrate[Exp[x Log[x]],x]

Integrate[Exp[x Log[x]],{x,0,1}]

Sum

Integrate[((((x Log[x]))^n)/(n!)),{x,0,1},Assumptions->n≥0]

NSum[(((-1))^n)(((1+n))^(-1-n)),{n,0,∞}]

NSum[((((-1))^(n+1))/(n^n)),{n,∞}]

(∫_0^1)(1/(x^x))dx

Integral

Integrate[(1/(x^x)),x]

Integrate[(1/(x^x)),{x,0,1}]

Integrate[Exp[-x Log[x]],x]

Integrate[Exp[-x Log[x]],{x,0,1}]

Sum

Integrate[((((-x Log[x]))^n)/(n!)),{x,0,1},Assumptions->n≥0]

NSum[(((1+n))^(-1-n)),{n,0,∞}]

NSum[(n^(-n)),{n,∞}]