Improper integration by truncation

For a nonnegative unbounded function, one truncation convention sets

f_{r} = min (f, r)

and asks that every

f_{r}

be Riemann integrable and that

\int f_{r}

have a finite limit as

r \to \infty

.

Table of contents
- First occurrence of a decimal digit Improper integration by truncation

First occurrence of a decimal digit

Under uniform length on

[0, 1]

K

of the first occurrence of a fixed decimal digit has

Pr (K = k) = \frac{1}{10} (\frac{9}{10})^{k - 1} .

The set of expansions in which the digit never occurs has length zero, and

E K = \sum_{k \geq 1} \frac{k}{10} (\frac{9}{10})^{k - 1} = 10.

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