kill(all)$

maxima:sum infinity ( = value ?),version 1.0,part 3
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)

packages  , option

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

Given : infinite sum

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

We need function : no faculty , denominator,
sin          → faculty
cos         →faculty
log(x+1)   →no faculty? check: with maxima
 -->    /*lhs(p3a1)=simplify_sum(lhs(p3a1))*/;
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); 

1e) step : highest factor,'n+2' : numerator (need differentation)

qdif(expr,xx):=diff(expr,xx);

p1a:qdif(x^(n+2),x);

p1a1:qdif(p1a,x);

2e) step: combination faculty (denominator=exp(x)) & x^m

→ use q4a

p2:intosum(x^m*q4a);

→ from : p1a,follows m=2

p2a:subst([m=2],p2);

rem :differentation no constant ,needed (only : with integration)

3e) step: differentation of expr 'p2a'

p3:'qdif(p2a,x);

p3a:ev(p3,nouns);

4e) step: differentation of expr 'p3a'

p4:'qdif(p3a,x);

p4a:ev(p4,nouns);

p4b:reverse(p4a);

p4c:subst([i=n],p4b);

5e) step: set x=1/3

p5:subst([x=1/3],p4c);

p5a:float(p5);

? check: with maxima

p6:lhs(p5)=simplify_sum(lhs(p5));
p6a:float(p6);