# Infinite sum smoothness Proof

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:

We know that if

$\forall i$ $f_i() \in C^{1}$

$s(n)=\sum_{i=1}^\infty f_i(n)< \infty$ (not neceesary uniformly)

and

$\sum_{i=1}^\infty f_i^{'}(n)=\sigma(n)$ uniformly

then:

$s(n) \in C^{1}$ and $s^{'}(n)=\sigma(n)$

So IN MY CASE we have $C^{\infty}$

$s(n)=\sum_{i=1}^\infty f(b_i(n))b_i^{'}(n)- f(a_i(n))a_i^{'}(n)< \infty$

and we need to show that:

$\sum_{i=1}^\infty h_i^{'}(n)$ converges uniformly,

where $h_i(n)=f(b_i(n))b_i^{'}(n)- f(a_i(n))a_i^{'}(n)$

MY QUESTION: Can somebody help me with any ideas how I can prove that $\sum_{i=1}^\infty h_i^{'}(n)$ converges uniformly.

I will really appreciate your help!

If we don't place any further restrictions on $$f,b_i$$ and $$a_i$$, then it's not true that $$\sum h_i(n)$$ converges uniformly.
Let $$\{U_i\}$$ be a partition of unity on $$\Bbb R$$, and let $$H_i$$ be an antiderivative of $$U_i$$. We want to show that there exist smooth functions $$f, b_i$$ and $$a_i$$ such that $$H_i(n)=f(b_i(n))b_i^{'}(n)- f(a_i(n))a_i^{'}(n)$$. This is easily achieved: take $$f\equiv 1$$ and $$a_i\equiv 0$$ to be constant functions, and $$b_i$$ an antiderivative of $$H_i$$.
By definition, $$\displaystyle\sum_{i=1}^\infty H_i(n) = \sum_{i=1}^\infty U_i(n) = 1$$ for all $$n$$. However, this convergence cannot be uniform because the support of each $$U_i$$ is compact, hence the support of any partial sum is compact. In particular, for any $$K$$ there is always some number $$n_0$$ such that $$\displaystyle \sum_{i=1}^K U_i(n_0) = 0$$, and this is automatically an obstruction to uniform convergence.
• Thank you a lot for your answer! Can you please propose restrictions, which should be applied to $f_i$,$b_i$ and $a_i$ to achieve $\sum_{i=1}^\infty h_i^{'}(n)$ converges uniformly? – Caim Feb 18 at 13:16