Consider the sequence $f_n(x)=e^{-nx}-2e^{-2nx}$ for $x>0$
a)Show $f(x):=\sum^\infty_{n=1}f_n(x)$ is well defined for $x>0$ and determine $f$
b) Show $$\sum^\infty \int_{(0,\infty)}f_n(x)\neq \int_{(0,\infty)}\sum^\infty f_n(x)$$
I dont even know how to attack this question, For part (a) it seems like if I have $x\neq y$ and $x,y>0$ the function is trivially well defined but I dont know what does the question mean by determine the function. for part (b) I was thinking about using Fubini's theorem and use the fact that we can see the sum as integrating with respect to the counting measure, let $\mu$ be counting measure on $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ then we have $$\sum^\infty \int_{(0,\infty)}f_n(x)=\int_\mathbb{N}\int_{(0,\infty)}f(n,x)dxd\mu$$ and $$ \int_{(0,\infty)}\sum^\infty f_n(x)=\int_{(0,\infty)}\int_\mathbb{N}f(n,x)d\mu dx$$ And since $\mu$ is not sigma finite we must not have the above two equal.
Can someone give me some idea on how to get started ? thanks