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

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)

• Well, you'd have to define your uncountable sum. In any case, the examples you give are countable.
– lulu
Mar 18 at 21:02
• Please edit your post for clarity. As it stands, it's really not clear what you are asking.
– lulu
Mar 18 at 21:03
• I'm wondering whether you truly intend "uncountable", or, rather, perhaps merely "infinite" as opposed to "finite"? Can you clarify? Mar 18 at 21:08
• @lulu sorry. i correct it. Mar 18 at 21:12
• @paulgarrett sorry . i correct it . i mean infinite sum. Mar 18 at 21:15

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.