kill(all)$

maxima:sum infinity ( = value ?),version 1.0,part 5
peter.vlasschaert@gmail.com,08/12/2020

see:maxima:sum infinity ( = value ?),version 1.0,part 1 (→ integration x1)
       maxima:sum infinity ( = value ?),version 1.0,part 2 (→ integration x2)
       maxima:sum infinity ( = value ?),version 1.0,part 3 (→ differentation x2)
       maxima:sum infinity ( = value ?),version 1.0,part 4 (→ differentation x1,integration x1)

packages  , option

load(simplify_sum)$
load(lrats)$ratprint:false$

Given : infinite sum

p1:S='sum((-1)^n*((%pi)^(2*n-1)*(2*n+3))/((n+1)*16^(n)*(2*n)!),n,0,∞);

p11:lhs(p1)=intosum(rhs(p1));

We need function :  faculty , denominator,
------------------------------------------------------------------------------------
sin          → faculty
cos         →faculty       (good,even,alternating series,see :p11)
log(x+1)   →no faculty
exp(x)      →faculty
------------------------------------------------------------------------------------

use : menu:Calculus:Get Series

float(%e^(1));
float(%pi);

1e) step

q1:niceindices(powerseries(sin(x), x, 0))$
q1a:sin(x) =q1;
q2:niceindices(powerseries(cos(x), x, 0))$
q2a:cos(x) =q2;
q3:niceindices(powerseries(log(1-x), x, 0))$
q3a:(log(1-x) =q3); /*math: ln(1-x)=log(1-x) */
q4:niceindices(powerseries(exp(x), x, 0))$
q4a:(exp(x) =q4); 

1e)   step : use cos(x), see above(good)
------------------------------------------------------

p1a:q2a;

p1a1:subst([i=n],p1a);

2e) step : integration :(n+1),denominator
-----------------------------------------------------
something: cos(x)

→ info: integration

assume(n>0);/*forget,see other parts*/

p2:'integrate(x^(2*n)*x,x);

p2a:factor(ev(p2,nouns));

end : info

→ see :p2 ,use integration constant = c1

p2a1:'integrate(x*p1a1,x);

p2a2:intosum(ev(p2a1,nouns));

p2a3:ev(p2a2,nouns);

→ find : %c1, set x=0

p2a4:reverse(subst([x=0],p2a3));

p2a5:subst([p2a4],p2a3);

p2a6:p2a5+rhs(p2a4)*(-1);

p2a7:reverse(p2a6);

3e) step : differentation :(2n+3),denominator
---------------------------------------------------------

→ info: differentation:(2n+3),numerator

p3:'diff(x^(2*n+2)*x,x);

p3a:ev(p3,nouns);

end:info

p3a1:'diff(intosum(x*p2a7),x);

p3a2:ev(p3a1,nouns);

4e) step : set x=π/4
----------------------------

p4:subst([x=%pi/4],p3a2);

→ representation

p4a:args(rhs(p11))=args(lhs(p4));

p4a1:lhs(p4a)[1];
p4a2:rhs(p4a)[1];
p4a3:ratsimp(p4a1/p4a2);

p4a4:expand((2*4^(2*n+2))/(%pi^3*16^n))=(2*4^2)/%pi^3;

p4a5:rhs(p11)=rhs(p4a4)*rhs(p4);

p4a6:float(p4a5);

load(hypergeometric)$

? check: with maxima

p5:rhs(p11)=ev(simplify_sum(rhs(p11)),nouns);

Hypergeometric series :
==================

Function: %f [p,q] ([a],[b],z)
------------------------------------
→  The pFq(a1,a2,..ap;b1,b2,..bq;z) hypergeometric function, 
      where a a list of length p and b a list of length q. 

p5a:lhs(p5) = hypergeometric_simp(float(rhs(p5)));

 →  How to use : Hypergeometric function ( output :from Maxima)

appendix :  
========

 ======================================================

(%i1) load ("hypergeometric") $
(%i2) foo : [hypergeometric([1,1], [2], z), hypergeometric([1/2], [1], z)];
(%o2) [hypergeometric([1, 1], [2], z), 
                                                     1
                                     hypergeometric([-], [1], z)]
                                                     2
(%i3) hypergeometric_simp (foo);
                 log(1 - z)              z    z/2
(%o3)         [- ----------, bessel_i(0, -) %e   ]
                     z                   2
(%i4) bar : hypergeometric([n], [m], z + 1);
(%o4)            hypergeometric([n], [m], z + 1)
(%i5) hypergeometric_simp (bar);
(%o5)            hypergeometric([n], [m], z + 1)

 ======================================================

ref:http://maxima.sourceforge.net/docs/manual/maxima_87.html