The most common command for numerical summation is evalf(Sum(f, x=a..b)) where the summation command is expressed in inert form to avoid first invoking the symbolic summation routines. It is also possible to invoke evalf on an unevaluated sum returned by the symbolic sum command, as in evalf(sum(f, x=a..b)), if it happens that symbolic sum fails (returns an unevaluated sum).

You can enter the command evalf/Sum using either the 1-D or 2-D calling sequence. For example, evalf(Sum(1/(x^2+1), x=-5..5)) is equivalent to $\mathrm{evalf}\left(\sum _{x=\mathrm{-5}}^5\frac{1}{x^2+1}\right)$

The evalf(Sum()) command attempts to compute a result that is accurate to within 1 ulp at the current setting of Digits, so increasing Digits increases the requested accuracy of the result.

The summand f may be another unevaluated sum or integral, that is, nested forms are supported.

$\mathrm{Sum}\left(\exp \left(x\right)\,x=\mathrm{RootOf}\left({\mathrm{\_Z}}^5+\mathrm{\_Z}+1\right)\right)$

$\sum _{x=\mathrm{RootOf}\left({\mathrm{\_Z}}^5+\mathrm{\_Z}+1\right)}{\ⅇ}^x$

$\mathrm{evalf}\left(\right)$

$4.791792042+0.\mathrm{I}$

$\mathrm{evalf}\left(\mathrm{Sum}\left(\frac{1}{x^2+1}\,x=-5..5\right)\right)$

$2.794570136$

$\mathrm{evalf}\left(\mathrm{Sum}\left(\frac{1}{x^2+1}\,x=-20..20\right)\right)$

$3.055883631$

$\mathrm{evalf}\left(\mathrm{Sum}\left(\frac{1}{x^2+1}\,x=-100..100\right)\right)$

$3.133448418$

$\mathrm{evalf}\left(\mathrm{Sum}\left(\frac{1}{x^2+1}\,x=-\mathrm{\infty }..\mathrm{\infty }\right)\right)$

$3.153348095$

$\mathrm{xpr}≔\mathrm{Int}\left({\mathrm{BesselK}\left(0\,\mathrm{sqrt}\left({\left(2n+1\right)}^2+x^2\right)\right)}^2\,x=-\mathrm{\infty }..\mathrm{\infty }\right)$

$\mathrm{xpr}≔{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{BesselK}\left(0\,\sqrt{{\left(2n+1\right)}^2+x^2}\right)}^2\ⅆx$

$\mathrm{evalf}\left(\mathrm{Sum}\left(\mathrm{xpr}\,n=0..\mathrm{\infty }\right)\right)$

$0.3088174675$

$\mathrm{evalf}\left(\mathrm{Sum}\left(\frac{1}{\mathrm{sqrt}\left(x\right)}\,x=1..10\right)\right)$

$5.020997899$

$\mathrm{evalf}\left(\mathrm{Sum}\left(\frac{1}{\mathrm{sqrt}\left(x\right)}\,x=1..100\right)\right)$

$18.58960382$

$\mathrm{evalf}\left(\mathrm{Sum}\left(\frac{1}{\mathrm{sqrt}\left(x\right)}\,x=1..1000\right)\right)$

$61.80100877$

$\mathrm{evalf}\left(\mathrm{Sum}\left(\frac{1}{\mathrm{sqrt}\left(x\right)}\,x=1..\mathrm{\infty }\right)\right)$

$\mathrm{-1.460354509}$

$\mathrm{sum}\left(\frac{1}{\mathrm{sqrt}\left(x\right)}\,x=1..\mathrm{\infty }\right)$

$\mathrm{\infty }$

$\mathrm{\zeta }\left(z\right)=\mathrm{Sum}\left(\frac{1}{i^z}\,i=1..\mathrm{\infty }\right)$

$\mathrm{\zeta }\left(z\right)=\sum _{i=1}^{\mathrm{\infty }}\frac{1}{i^z}$

$\mathrm{evalf}\left(\mathrm{eval}\left(\,z=\frac{1}{2}\right)\right)$

$\mathrm{-1.460354509}=\mathrm{-1.460354509}$

$\mathrm{\_EnvFormal}≔\mathrm{true}\:$

$\mathrm{sum}\left(\frac{1}{\mathrm{sqrt}\left(x\right)}\,x=1..\mathrm{\infty }\right)$

$\mathrm{\zeta }\left(\frac{1}{2}\right)$

$\mathrm{forget}\left(\mathrm{evalf}\right)\:$$\mathrm{evalf}\left(\mathrm{Sum}\left(\frac{1}{\mathrm{sqrt}\left(x\right)}\,x=1..\mathrm{\infty }\right)\right)$

$\mathrm{-1.460354509}$

forget(evalf):
evalf(Sum(1/sqrt(x), x=1..infinity, formal=false));

$Float\left(\mathrm{\infty }\right)$

$\mathrm{\_EnvFormal}≔\mathrm{false}\:$

$\mathrm{forget}\left(\mathrm{evalf}\right)\:$$\mathrm{evalf}\left(\mathrm{Sum}\left(\frac{1}{\mathrm{sqrt}\left(x\right)}\,x=1..\mathrm{\infty }\right)\right)$

$Float\left(\mathrm{\infty }\right)$

$\mathrm{\_EnvFormal}≔'\mathrm{\_EnvFormal}'\:$

Fessler, T.; Ford, W.F.; and Smith, D.A. "HURRY: An acceleration algorithm for scalar sequences and series." ACM Trans. Math. Software, Vol. 9, (1983): 346-354.

Levin, D. "Development of non-linear transformations for improving convergence of sequences". Internat. J. Comput. Math, Vol. B3, (1973): 371-388.

The c parameter was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.