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

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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)

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