For every x ∈ [0, 1] and n ∈ N, define fn(x) to be the distance between x and the nearest point in [0, 1] of the form 10^−nk, where k is an integer. If f(x) = ∑n=1^∞ fn(x), evaluate ∫[0,1] f dm.

Question

For every x ∈ [0, 1] and [latex]n ∈ \mathbb{N},[/latex] define [latex]f_{n}(x)[/latex] to be the distance between x and the nearest point in [0, 1] of the form [latex]10^{−n}k,[/latex] where k is an integer. If [latex]f (x) = \Sigma_{n=1}^{∞}{f_{n}(x)},[/latex] evaluate [latex]\int_{[0,1]}{f} dm.[/latex]

Accepted Answer

Taking [latex]x ∈ [10^{−n}k, 10^{−n}(k +1))[/latex] with [latex]0 ≤ k ≤ 10^{n},[/latex] we have

[latex]f_{n}(x) = \begin{cases} x − 10^{−n}k, & x ∈ I_{n}^{k} \ 10^{−n}(k + 1) − x, &x ∈ J_{n}^{k} , \end{cases}[/latex]

where

[latex]I_{n}^{k} = \left[\frac{k}{10^{n}}, \frac{2k + 1}{2 × 10^{n}} ight) , J_{n}^{k} = \left[\frac{2k + 1}{2 × 10^{n}}, \frac{k + 1}{10^{n}} ight).[/latex]

Using the result of Example 11.1 and the linearity of the integral, we obtain

[latex]\int_{I_{n}^{k}}{f_{n}} dm =\int_{J_{n}^{k}}{f_{n}} dm = \frac{1}{8 × 10^{2n}}[/latex]

[latex]⇒ \int_{[0,1]}{f_{n}} dm = \sum\limits_{k=0}^{10^{n}−1}{\int_{I_{n}^{k}∪J_{n}^{k}}{f_{n}} dm = \frac{1}{4 × 10^{n}}} .[/latex]

Since [latex]f_{n} ≥ 0,[/latex] Corollary 11.3.1 gives

[latex]\int_{[0,1]}{f} dm = \sum\limits_{n=1}^{∞}{\frac{1}{4 × 10^{n}}}= \frac{1}{36}.[/latex]

Question 11.4: For every x ∈ [0, 1] and n ∈ N, define fn(x) to be the dista...

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