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