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

Question

i know that if $f_1(x),f_2(x),...,f_n(x)$ is odd/even functions and $c_1,c_2,...,c_n$ are fixed real numbers then $c_1 f_1(x) + c_2 f_2(x) +... +c_n f_n(x)$ 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)

Answer 1

Yes. Let $(f_n)$ be a sequence of even functions, $(c_n)$ a sequence of real numbers. Let $f = \sum _{n = 1} ^\infty c_n f_n$. Define $S_n = \sum _{k = 1} ^n c_kf_k$, then $S_n \to f$ as $n\to \infty$. $S_n$ is even for all $n$, as for all x $S_n(x) = c_1f_1(x) + … + c_nf_n(x) = c_1f_1(-x) + … + c_nf_n(-x) = S_n(-x)$. Therefore, for all x $f(x) = \lim S_n(x) = \lim S_n(-x) = f(-x)$, which means $f$ is even. The proof in the case of odd functions is very similar.