Function: suminf
Section: sums
C-Name: suminf0
Prototype: V=GEb
Help: suminf(X=a,expr): naive summation (X goes from a to infinity) of real or
 complex expression expr.
Wrapper: (,G)
Description:
  (gen,gen):gen:prec suminf(${2 cookie}, ${2 wrapper}, $1, $prec)
Doc: Naive summation of expression \var{expr}, the formal parameter $X$
 going from $a$ to infinity. The evaluation stops when the relative error of
 the expression is less than the default bit precision for 3 consecutive
 evaluations. The expressions must evaluate to a complex number.

 If the expression tends slowly to $0$, like $n^{-a}$ for some $a > 1$,
 make sure $b = \kbd{realbitprecision}$ is low: indeed, the algorithm will
 require $O(2^{b/a})$ function evaluations and we expect only about $b(1-1/a)$
 correct bits in the answer. If the series is alternating, we can expect $b$
 correct bits but the \tet{sumalt} function should be used instead since its
 complexity is polynomial in $b$, instead of exponential. More generally,
 \kbd{sumpos} should be used if the terms have a constant sign and
 \kbd{sumnum} if the function is $C^{\infty}$.

 \bprog
 ? \pb25
   realbitprecision = 25 significant bits (7 decimal digits displayed)
 ? exponent(suminf(i = 1, (-1)^i / i) + log(2))
 time = 2min, 2,602 ms.
 %1 = -29
 ? \pb45
   realbitprecision = 45 significant bits (13 decimal digits displayed)
 ? exponent(suminf(i = 1, 1 / i^2) - zeta(2))
 time = 2,186 ms.
 %2 = -23

 \\ alternatives are much faster
 ? \pb 10000
   realbitprecision = 10000 significant bits (3010 decimal digits displayed)
 ? exponent(sumalt(i = 1, (-1)^i / i) + log(2))
 time = 25 ms.
 %3 = -10043

 ? \pb 4000
   realbitprecision = 4000 significant bits (1204 decimal digits displayed)))
 ? exponent(sumpos(i = 1, 1 / i^2) - zeta(2))
 time = 22,593 ms.
 %4 = -4030

 ? exponent(sumnum(i = 1, 1 / i^2) - zeta(2))
 time = 7,032 ms.
 %5 = -4031

 \\ but suminf is perfect for geometrically converging series
 ? exponent(suminf(i = 1, 2^-i) - 1)
 time = 25 ms.
 %6 = -4003
 @eprog

 \synt{suminf}{void *E, GEN (*eval)(void*,GEN), GEN a, long prec}.
