prove or disprove that "the sum of infinite even or odd functions is even or odd".

prove or disprove that "the sum of infinite even or odd functions is even or odd".

i know that if f1(x),f2(x),...,fn(x) is odd/even functions and c1,c2,...,cn are fixed real numbers then c1f1(x)+c2f2(x)+...+cnfn(x) is odd/even.
But how for infinite sum ((sum of (fi , i member of I))?! exactly i want to prove or disprove that "the sum of infinite even or odd function is even or odd". i think its wrong but not sure !!! i have an example in my mind but i dont know its correct or not!!

let f1(x)=1 , f2(x)=-1 , f3(x)=1 , f4(x)=-1 ,....
all of fi(x) are even. but is f1+f2+f3+... = 1-1+1-1+... even function?

and let g1(x)=x , g2(x)=-x , g3(x)=x , g4(x)=-x ,....
all of gi(x) are odd. but is g1+g2+g3+... = x-x+x-x+... odd function?

i dont know these examples are correct or not.(if correct how to continue my proof exactly and if not please prove its odd/ even)