Function: sumpos
Section: sums
C-Name: sumpos0
Prototype: V=GED0,L,p
Help: sumpos(X=a,expr,{flag=0}): sum of positive (or negative) series expr,
 the formal
 variable X starting at a. flag is optional, and can be 0: default, or 1:
 uses a slightly different method using Zagier's polynomials.
Wrapper: (,G)
Description:
  (gen,gen,?0):gen:prec sumpos(${2 cookie}, ${2 wrapper}, $1, $prec)
  (gen,gen,1):gen:prec sumpos2(${2 cookie}, ${2 wrapper}, $1, $prec)
Doc: numerical summation of the series \var{expr}, which must be a series of
 terms having the same sign, the formal variable $X$ starting at $a$. The
 algorithm uses Van Wijngaarden's trick for converting such a series into
 an alternating one, then \tet{sumalt}. For regular functions, the
 function \kbd{sumnum} is in general much faster once the initializations
 have been made using \kbd{sumnuminit}. Contrary to \kbd{sumnum},
 \kbd{sumpos} allows functions defined only at integers:
 \bprog
 ? sumnum(n = 0, 1/n!)
  ***   at top-level: sumnum(n=1,1/n!)
  ***                              ^---
  ***   incorrect type in gtos [integer expected] (t_FRAC).
 ? sumpos(n = 0, 1/n!) - exp(1)
 %2 = -1.0862155548773347717 E-33
 @eprog\noindent On the other hand, when the function accepts general real
 numbers, it is usually advantageous to replace $n$ by \kbd{$n$ * 1.0} in the
 sumpos call in particular when rational functions are involved:
 \bprog
 ? \p500
 ? sumpos(n = 0, n^7 / (n^9+n+1));
 time = 6,108 ms.
 ? sumpos(n = 0, n *= 1.; n^7 / (n^9+n+1));
 time = 2,788 ms.
 ? sumnumrat(n^7 / (n^9+n+1), 0);
 time = 4 ms.
 @eprog\noindent In the last example, \kbd{sumnumrat} is of course much
 faster but it only applies to rational functions.

 The routine is heuristic and assumes that \var{expr} is more or less a
 decreasing function of $X$. In particular, the result will be completely
 wrong if \var{expr} is 0 too often. We do not check either that all terms
 have the same sign: as \tet{sumalt}, this function should be used to
 try and guess the value of an infinite sum.

 If $\fl=1$, use \kbd{sumalt}$(,1)$ instead of \kbd{sumalt}$(,0)$, see
 \secref{se:sumalt}. Requiring more stringent analytic properties for
 rigorous use, but allowing to compute fewer series terms.

 To reach accuracy $10^{-p}$, both algorithms require $O(p^{2})$ space;
 furthermore, assuming the terms decrease polynomially (in $O(n^{-C})$), both
 need to compute $O(p^{2})$ terms. The \kbd{sumpos}$(,1)$ variant has a smaller
 implied constant (roughly 1.5 times smaller). Since the \kbd{sumalt}$(,1)$
 overhead is now small compared to the time needed to compute series terms,
 this last variant should be about 1.5 faster. On the other hand, the
 achieved accuracy may be much worse: as for \tet{sumalt}, since
 conditions for rigorous use are hard to check, the routine is best used
 heuristically.

 \synt{sumpos}{void *E, GEN (*eval)(void*,GEN),GEN a,long prec}. Also
 available is \tet{sumpos2} with the same arguments ($\fl = 1$).
