# proof of Ruffa’s formula for continuous functions

Define $s_{n}$ to be the following sum:

 $s_{n}=\sum\limits_{m=1}^{2^{n}-1}2^{-n}f\left({a+m(b-a)/2^{n}}\right)$

Making the substitution $m^{\prime}=2m$ and using the fact that $1+(-1)^{m^{\prime}}=0$ when $m^{\prime}$ is odd to express the sum over even values of $m^{\prime}$ as a sum over all values of $m^{\prime}$, this becomes

 $s_{n}=\sum\limits_{m=1}^{2^{n+1}-1}(1+(-1)^{m^{\prime}})2^{-1-n}f\left({a+m(b-% a)/2^{n}}\right)$

Subtracting this sum from $s_{n+1}$ and simplifying gives

 $s_{n+1}-s_{n}=\sum\limits_{m=1}^{2^{n+1}-1}(-1)^{m+1}2^{-n}f\left({a+m(b-a)/2^% {n}}\right)$

Using the telescoping sum trick, we may write

 $s_{k}=\sum_{n=1}^{k}(s_{n}-s_{n-1})=\sum\limits_{n=1}^{k}{\sum\limits_{m=1}^{2% ^{n}-1}{\left({-1}\right)^{m+1}}}2^{-n}f\left({a+m(b-a)/2^{n}}\right)$

To complete the proof, we must investigate the limit as $k\to\infty$. Since $f$ is assumed continuous and the interval $[a,b]$ is compact, $f$ is uniformly continuous. This means that, for every $\epsilon>0$, there exists a $\delta>0$ such that $|x-y|<\delta$ implies $|f(x)-f(y)|<\epsilon$. By the Archimedean property, there exists an integer $k>0$ such that $2^{k}\delta>|a-b|$. Hence, $|f(x)-f\left({a+m(b-a)/2^{n}}\right)|\leq\epsilon$ when $x$ lies in the interval $[a+(m-1)(b-a)/2^{n},a+(m+1)(b-a)/2^{n}]$. Thus, $(a-b)s_{k}+|a-b|\epsilon$ is a Darboux upper sum for the integral

 $\int_{a}^{b}f(x)\,dx$

and $(b-a)s_{k}-|a-b|\epsilon$ is a Darboux lower sum. (Darboux’s definition of the integral may be thought of as a modern incarnation of the ancient method of exhaustion.) Hence

 $|\int_{a}^{b}f(x)\,dx-s_{k}|\leq|a-b|\epsilon$

Taking the limit $\epsilon\to 0$, we see that

 $\int\limits_{a}^{b}{f(x)dx=\sum\limits_{n=1}^{\infty}{A_{n}}=\left({b-a}\right% )}\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{2^{n}-1}{\left({-1}\right)^{m+% 1}}}2^{-n}f\left({a+m(b-a)/2^{n}}\right)$
Title proof of Ruffa’s formula for continuous functions ProofOfRuffasFormulaForContinuousFunctions 2013-03-22 14:56:41 2013-03-22 14:56:41 rspuzio (6075) rspuzio (6075) 8 rspuzio (6075) Proof msc 30B99 msc 26B15 msc 78A45