djc

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17 years, 216 days
Technical professional in industry or government
Budapest, Hungary

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These are answers submitted by djc

Using the command csum (from the Maple Advisor Database by Robert Israel) the convergence of the summation can be checked:

restart;

a := 2/sqrt(n+2)-2/sqrt(n+3);

csum(a,n);

true

# in Maple 12:

restart;

with(SumTools):

_EnvFormal := true;

a := 2/sqrt(n+2)-2/sqrt(n+3);

Summation(a, n=1..infinity );

2/3*3^(1/2)

You can also check the positivity of a function with the command verify, but it doesn't give you the points where the function is positive.
See below:

restart;
assume(x,'real');
verify(x^2+1,0,'greater_than' ); # it gives true, i.e. the function x^2+1 is positive for every real x.
 

verify(x^2,0,'greater_than' );  # it gives FAIL: the function x^2 is not everywhere positive (or in another case Maple can not decide it.)

verify(-x^2-1,0,'greater_than' )  ;# it gives false: the function -x^2-1 is everywhere negative.
 

 verify(sin(x),0,'greater_than' ) assuming x>0 and x<Pi;  # it gives true:   sin(x) is positive on the range of (0,Pi).

verify(sin(x),0,'greater_than' ) assuming x>Pi and x<2*Pi; # it gives false:  sin(x) is negative on the range of (Pi,2*Pi)

Try this:

F := t -> 1+x(t)^2+t^3 ;
D[1](F)(1);
algsubs(x(1)=-4,algsubs(D(x)(1)=F(1),%));

Try this:

simplify(sqrt( x^(2*a+2))) assuming x::positive,a::real;

Or without assumpiton with the symbolic keyword:

simplify(sqrt(  x^(2*a+2) ),symbolic);

 

I have changed this row in your code to get a simplified formula:

seq(1/Pi*int(f*F(n*x),x=-Pi..Pi),F in [cos,sin]) assuming n::posint
 

a0,an,bn := Fouriercoeff(x^2);

a0, an, bn := 2/3*Pi^2, 4/n^2*(-1)^n, 0

If y is your expression: var:=op(op(selectfun(y,arctan)));
Use exp(1) instead of e: f := x->exp(1)^(-x)*sin(x); plot(f(x),x=0..2*Pi);
test:=proc(h) unapply(h,x)(1.1) end proc; test(x^3); # Another solution: test:=proc(h) eval(h,op(indets(h))=1.1) end proc; test(t^3);
You can try this: restart; f := x[1]+x[1]*x[2]+x[1]*x[2]*x[3]; diff(f, 'x[1]'); int(int(int(f, 'x[1]' = 0 .. 1), 'x[2]' = 0 .. 1), 'x[3]' = 0 .. 1);
It is nice to see that my problem can be directly solved in maple 11. Good work!
It seems to me this bug is fixed in maple 11. Maple 11 gives the full solution. Thanks.
I am happy to see that maple 11 is definitely stronger than maple 10. I can calculate the mean and the standard deviation of my original expression with maple 11 symbolically.
You can check the convergence of series with the csum command of the freely downloadable Advisor package. http://www.math.ubc.ca/~israel/advisor/ csum(((Pi+3*cos(x))/x, x )); false
You can also try verify: restart; myrange:= 1..10; verify( x, myrange, 'interval'('closed') ); # FAIL x:=1; verify( x,myrange, 'interval'('closed') ); # true verify( x, myrange, 'interval'('open') ); # fail
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