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Edit detail for SandBoxSymbolicSumsAndProducts revision 2 of 2

1 2
Editor: Bill Page
Time: 2023/10/23 16:17:38 GMT+0
Note:

changed:
-numeric %
%::Expression Float

Symbolic evaluation of sums and products

fricas
(1) -> sum(i, i=1..n)

\label{eq1}\frac{{{n}^{2}}+ n}{2}(1)
Type: Fraction(Polynomial(Integer))
fricas
product(x+k, k=0..n-1)

\label{eq2}\prod_{
\displaystyle
{k = 0}}^{
\displaystyle
{n - 1}}{\left(x + k \right)}(2)
Type: Expression(Integer)

Result should be pi

fricas
integrate((4*sqrt(1/2)-8*x^3-4*sqrt(1/2)*x^4-8*x^5)/(1-x^8), x=0..sqrt(1/2))

\label{eq3}\verb#"potentialPole"#(3)
Type: Union(pole: potentialPole,...)

Use "noPole"

Axioms answer is "potentialPole", which indicates that there might be a pole within the interval of integration. If you are sure, that there is no pole within this interval, use "noPole":

fricas
integrate((4*sqrt(1/2)-8*x^3-4*sqrt(1/2)*x^4-8*x^5)/(1-x^8), x=0..sqrt(1/2), "noPole")

\label{eq4}-{{\frac{1}{2}}\ {\log \left({\frac{5}{2}}\right)}}+{{\frac{1}{4}}\ {\log \left({\frac{1}{4}}\right)}}-{\arctan \left({2}\right)}+ \pi(4)
Type: Union(f1: OrderedCompletion?(Expression(AlgebraicNumber?)),...)

It's numeric value is roughly

fricas
%::Expression Float

\label{eq5}1.2297249795 \<u> 786525481(5)
Type: Expression(Float)

To check, do a numeric integration:

fricas
romberg(x+->(4*sqrt(1/2)-8*x^3-4*sqrt(1/2)*x^4-8*x^5)/(1-x^8),0.0,sqrt(1/2)::Float,0.1,0.1,6,10)

\label{eq6}\begin{array}{@{}l}
\displaystyle
\left[{value ={1.2297249795 \_ 786311161}}, \: \right.
\
\
\displaystyle
\left.{error ={0.251271954 E - 10}}, \:{totalpts ={129}}, \:{success =  \mbox{\rm true} }\right] 
(6)
Type: Record(value: Float,error: Float,totalpts: Integer,success: Boolean)