Riemann Integrability

**namelessguy** · May 15th 2008, 01:03 PM

I have no idea how to prove this. Can someone help?
Given a real number x, denote by [x] the largest integer less than or equal to x. Prove that the function defined
$f(x)$ = $\sum_{n = 0}^{\infty}\frac{nx-[nx]}{2^n}$ is Riemann integrable on any interval [a,b]. Then compute
$\int_0^1 f(x)dx$

**NonCommAlg** · May 15th 2008, 03:11 PM

Originally Posted by namelessguy

I have no idea how to prove this. Can someone help?
Given a real number x, denote by [x] the largest integer less than or equal to x. Prove that the function defined
$f(x)$ = $\sum_{n = 0}^{\infty}\frac{nx-[nx]}{2^n}$ is Riemann integrable on any interval [a,b]. Then compute
$\int_0^1 f(x)dx$

for n = 0 the term in your series is 0. so i'll assume that n > 0. let $f_n(x)=\frac{nx - [nx]}{2^n}, \ n \geq 1.$

each $f_n$ is integrable on any interval [a, b], because it's continuous almost everywhere on the

interval. let $\sigma_n(x)=\sum_{j=1}^n f_j(x).$ so $\sigma_n$ is integrable on [a, b] for all n.

since $0 \leq f_n(x) \leq \frac{1}{2^n}, \ \forall x,$ by Weierstrass test, $\{\sigma_n \}$ is uniformly convergent. so $f(x)=\lim_{n\to\infty} \sigma_n(x)$

is integrable on [a, b] and $\int_a^b f(x) dx = \lim_{n\to\infty} \int_a^b \sigma_n(x) dx.$ if a = 0 and b = 1, then this gives us:

$\int_0^1 f(x) dx = \lim_{n\to\infty} \int_0^1 \sigma_n(x) dx = \lim_{n\to\infty} \int_0^1 \sum_{j=1}^n f_j(x) \ dx=\lim_{n\to\infty} \sum_{j=1}^n \int_0^1 f_j(x) dx$

$=\lim_{n\to\infty} \sum_{j=1}^n \int_0^1 \frac{jx -[jx]}{2^j}dx=\lim_{n\to\infty} \sum_{j=1}^n \frac{1}{2^{j+1}}=\sum_{j=1}^{\infty} \frac{1}{2^{j+1}}=\frac{1}{2}. \ \ \ \square$

**ThePerfectHacker** · May 15th 2008, 03:22 PM

Originally Posted by namelessguy

I have no idea how to prove this. Can someone help?
Given a real number x, denote by [x] the largest integer less than or equal to x. Prove that the function defined
$f(x)$ = $\sum_{n = 0}^{\infty}\frac{nx-[nx]}{2^n}$ is Riemann integrable on any interval [a,b]. Then compute
$\int_0^1 f(x)dx$

I will do the special case $[a,b]=[0,1]$ . Define a sequence of functions $f_n:[0,1]\mapsto \mathbb{R}$ as $f_n(x) = \tfrac{1}{2^n}(nx-[nx])$ . Notice that $|f_n(x)| \leq \tfrac{1}{2^n}$ . Since $\sum _{n=0}^{\infty}\tfrac{1}{2^n} < \infty$ it follows by the Weierstrass test that the series of functions $\sum_{n=0}^{\infty} f_n(x)$ converges uniformly to a function $f(x)$ . Each function $f_n(x)$ is discontinous at $\tfrac{k}{n}$ for $1\leq k<n$ . This implies that $f$ is integrable and $\int_0^1 f(x) dx = \sum_{n=0}^{\infty}\int_0^1 f_n(x) dx$ . It remains to compute $\int_0^1 f_n(x) dx$ . The graph of $f_n$ consists of a $n$ congruent right-angles triangles each having height $\tfrac{1}{2^n}$ and with base width $1/n$ . Thus, $\int_0^1 f_n(x) dx = \frac{1}{2} n\cdot \frac{1}{n} \cdot \frac{1}{2^n} = \frac{1}{2^{n+1}}$ . I might have missed maybe $\pm 1$ on the exponent, but you get the idea. Now this is a geometric series which sums easily.

EDIT: Got beaten in the response by a fellow algebraist.

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