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inf-limit

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inf-limit-plot2d
plot2d([a[n]],[n,1,20]);

a[n]:=3+1/n;

limit(a[n],n,inf);

plot2d([a[n]],[n,1,20]);

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inf

Run Example
(%i1)integrate(log(x)/(x^2+(0.5)^2),x,0,inf);

rat: replaced 0.25 by 1/4 = 0.25

rat: replaced 0.25 by 1/4 = 0.25

rat: replaced 0.25 by 1/4 = 0.25

rat: replaced 0.25 by 1/4 = 0.25

rat: replaced 0.25 by 1/4 = 0.25
(%o1)                               4 false
(%i2) 
Run Example
f(x):=exp(-x^2);
                                              2
(%o1)                          f(x) := exp(- x )
(%i2) tex( expand( integrate( (x-y-l)*f(x), x, l+y, inf) ));
$${{\sqrt{\pi}\,y\,\mathrm{erf}\left(y+l\right)}\over{2}}+{{\sqrt{\pi
 }\,l\,\mathrm{erf}\left(y+l\right)}\over{2}}+{{e^{-y^2-2\,l\,y-l^2}
 }\over{2}}-{{\sqrt{\pi}\,y}\over{2}}-{{\sqrt{\pi}\,l}\over{2}}$$
(%o2)                                false
(%i3) tex(integrate( expand( integrate( (x-y-l)*f(x), x, l+y, inf) * f(y)),y, -inf,inf));
$$\int_{-\infty }^{\infty }{{{\sqrt{\pi}\,y\,e^ {- y^2 }\,
 \mathrm{erf}\left(y+l\right)}\over{2}}+{{\sqrt{\pi}\,l\,e^ {- y^2 }
 \,\mathrm{erf}\left(y+l\right)}\over{2}}-{{\sqrt{\pi}\,y\,e^ {- y^2
  }}\over{2}}-{{\sqrt{\pi}\,l\,e^ {- y^2 }}\over{2}}+{{e^{-2\,y^2-2\,
 l\,y-l^2}}\over{2}}\;dy}$$
(%o3)                                false
(%i4) integrate( expand( integrate( (x-y-l)*f(x), x, l+y, inf) * f(y)),y, -inf,inf);
       inf                     2                               2
      /                     - y                             - y
      [       sqrt(%pi) y %e     erf(y + l)   sqrt(%pi) l %e     erf(y + l)
(%o4) I      (----------------------------- + -----------------------------
      ]                     2                               2
      /
       - inf
                             2                    2          2            2
                          - y                  - y      - 2 y  - 2 l y - l
            sqrt(%pi) y %e       sqrt(%pi) l %e       %e
          - ------------------ - ------------------ + ---------------------) dy
                    2                    2                      2
(%i5) integrate( y*exp(-y^2), y, -inf,inf);
(%o5)                                  0
(%i6) integrate( exp(-y^2)*erf(y+l),y,-inf,inf );
                           inf
                          /           2
                          [        - y
(%o6)                     I      %e     erf(y + l) dy
                          ]
                          /
                           - inf
(%i7) integrate( y*exp(-y^2),y,-inf,inf);
(%o7)                                  0
(%i8) integrate( exp(-y^2),y,-inf,inf);
(%o8)                              sqrt(%pi)
(%i9) integrate( exp(-(y+l)^2),y,-inf,inf);
(%o9)                              sqrt(%pi)
(%i10) romberg(exp(-y^2)*erf(y+l),y,-inf,inf);
                              2
                           - y
(%o10)           romberg(%e     erf(y + l), y, - 1.0 inf, inf)
(%i11) 
Run Example
limit((243*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)           +(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^6           -18*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^5           -9*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^4           +270*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^3           -486*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^2)           *(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^(-i-1)           /(-10206*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)            +14*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^6            -18*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^5            -810*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^4            +1620*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^3            +3645*(3*sqrt(5)/4-3^(3/2)*sqrt(5-sqrt(5))/2^(3/2)-9/4)^2+5832),i,inf);
(%o1)                                  0
(%i2) 

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