kill(all)$

maxima:sum infinity ( = value ?),version 1.0,part 4
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)

packages  , option

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

Given : infinite sum

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

We need function :  faculty , denominator,
sin          → faculty
cos         →faculty
log(x+1)   →no faculty
exp(x)      →faculty
-------------------------------------------------------------

use : menu:Calculus:Get Series

1e) step

q1:niceindices(powerseries(sin(x), x, 0))$
q1a:sin(x) =q1;
q2:niceindices(powerseries(cos(x), x, 0))$
q2a:sin(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); 

what about : combination " integration" & "differentation"
---------------------------------------------------------------------------
when : quotient : above & below  :  example :1e)  ((n+1)*(n+2)*.....)/((n+3)*(n+4)*......) in S
                                                                     2e)   faculty : denominator ,see qai ,i=1,2,.....(above)

1e)  step : combination  x^m*exp(x)
-----------------------------------------------
use :q4a

p1a:intosum(x^m1*q4a);

2e) step : numerator , need : (n+3), differentation is a reduction of power: m1 =3
------------------------------------------------------------------------------------------------------------

p2a:subst([m1=3,i=n],p1a);

p2a1:'diff(p2a,x);

p2a2:ev(p2a1,diff);

3e) step : numerator , need : (n+2), integration is a increase of power: m2 =-1
------------------------------------------------------------------------------------------------------------
rem : denominator  1/x, need constant for integration = c1

p3a:x^m2*(p2a2);

p3a1:intosum(subst([m2=-1],p3a));

p3a2:'integrate(lhs(p3a1),x) = 'integrate(rhs(p3a1),x)+c1;

p3a3:ev(p3a2,nouns);

→ find c1: @x=0

p3a4:reverse(subst([x=0],p3a3));

p3a5:subst([p3a4],p3a3);

p3a6:p3a5+(rhs(p3a4)*(-1));

4e) step :x=1/2 ,for 2^n
-------------------------------

p4:subst([x=1/2],p3a6);

5e) step eliminate factor from rhs ,'p4a'
----------------------------------------------------
try equality for sum S & rhs

→ representation 

p5:args(rhs(p1)) = args(rhs(p4));

p5a:lhs(p5)[1];/*S*/

p5a1:rhs(p5)[1];/*result*/

→ ratio

p5a2:p5a1/p5a;

→  p4

p5a3:intosum(p4*1/p5a2); 

p5a4:reverse(distrib(p5a3));

p5a5:float(p5a4);

? check: with maxima

 p6:lhs(p5a4)=simplify_sum(lhs(p5a4));

p6a:float(p6);