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Question: Limitations of `sum`?

Question:Limitations of `sum`?

sursumCorda 1079 Maple
sum hypergeom
July 23 2023
0

Some work, while others don't work. 
 

> 

restart;
> 

`assuming`([is(1/(sqrt(n*(n+1))*(sqrt(n)+sqrt(n+1))) = 1/sqrt(n)-1/sqrt(n+1))], [n::posint])

true

 (1)
> 

sum(1/sqrt(n)-1/sqrt(n+1), n = 1 .. infinity)

1

 (2)
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sum(1/(sqrt(n*(n+1))*(sqrt(n)+sqrt(n+1))), n = 1 .. infinity)

sum(1/((n*(n+1))^(1/2)*(n^(1/2)+(n+1)^(1/2))), n = 1 .. infinity)

 (3)
> 

`assuming`([sum(binomial(3*n, n)*x^n/(2*n+1), n = 0 .. infinity)], [abs(x) <= 4/27])

(2/3)*3^(1/2)*sin((1/3)*arcsin((3/2)*3^(1/2)*x^(1/2)))/x^(1/2)

 (4)
> 

`assuming`([eval(binomial(3*n, n)*x^n/(2*n+1), n = 0)+sum(binomial(3*n, n)*x^n/(2*n+1), n = 1 .. infinity, parametric)], [abs(x) <= 4/27])

1+x*hypergeom([1, 4/3, 5/3], [2, 5/2], (27/4)*x)

 (5)
> 

simplify(convert(1+x*hypergeom([1, 4/3, 5/3], [2, 5/2], (27/4)*x)-hypergeom([1/3, 2/3], [3/2], (27/4)*x), elementary), symbolic)

(1/3)*(3*x^(1/2)+3*x^(3/2)*hypergeom([1, 4/3, 5/3], [2, 5/2], (27/4)*x)-2*3^(1/2)*sin((1/3)*arcsin((3/2)*3^(1/2)*x^(1/2))))/x^(1/2)

 (6)
> 

plot((1/3)*(3*x^(1/2)+3*x^(3/2)*hypergeom([1, 4/3, 5/3], [2, 5/2], (27/4)*x)-2*3^(1/2)*sin((1/3)*arcsin((3/2)*3^(1/2)*x^(1/2))))/x^(1/2), x = -4/27 .. 4/27)

 
> 

verify(0, (1/3)*(3*x^(1/2)+3*x^(3/2)*hypergeom([1, 4/3, 5/3], [2, 5/2], (27/4)*x)-2*3^(1/2)*sin((1/3)*arcsin((3/2)*3^(1/2)*x^(1/2))))/x^(1/2), 'equal')

FAIL

 (7)
> 

NULL


 

Download unable_to_sum.mw

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