There is a function:
$V(n) = \sum_{i=1}^\infty f(b_i(n))b_i^{'}(n)- f(a_i(n))a_i^{'}(n)$,
where $f$,$b$ and $a$ are $C^\infty$ smooth functions.
I know that we can prove that the function $V(n)$ is also smooth under some assumptions.
I SUGGEST THE NEXT SOLUTION
As I understand we should take
$V^{'}(n) = \sum_{i=1}^\infty f^{'}(b_i(n))b_i^{'}(n)^2+f(b_i(n))b_i^{''}(n)- f^{'}(a_i(n))a_i^{'}(n)^2-f(a_i(n))a_i^{''}(n)$,
where $f^{'}$ exists, $b^{''}$ and $a^{''}$ should also be smooth.
MY QUESTION: Can somebody check this logic and help me?
I will really appreciate your help!