# Sum of Reciprocals in Base 10 with Zeroes Removed

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## Theorem

The infinite series

- $\ds \sum_{\map P n} \frac 1 n$

where $\map P n$ is the propositional function:

- $\forall n \in \Z_{>0}: \map P n \iff$ the decimal representation of $n$ contains no instances of the digit $0$

converges to the approximate limit $23 \cdotp 10345 \ldots$

## Proof

For each $k \in \N$, there are $9^k$ $k$-digit numbers containing no instances of the digit $0$.

Each of these numbers is at least $10^{k - 1}$.

Hence the reciprocals of each of these numbers is at most $\dfrac 1 {10^{k - 1}}$.

Thus:

\(\ds \sum_{\map P n} \frac 1 n\) | \(<\) | \(\ds \sum_{k \mathop = 1}^\infty \frac {9^k} {10^{k - 1} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac 9 {1 - \frac 9 {10} }\) | Sum of Geometric Sequence | |||||||||||

\(\ds \) | \(=\) | \(\ds 90\) |

showing that the sum converges.

This needs considerable tedious hard slog to complete it.In particular: Finer approximations can be obtained (e.g. in virtue of Closed Form for Triangular Numbers/Direct Proof), but due to the slow growth of the harmonic series, many numbers must to summed to obtain the approximation in the theorem. Case in point: $H_{5 \times 10^9} < 23$To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- Oct. 1971: R.P. Boas, Jr. and J.W. Wrench, Jr.:
*Partial Sums of the Harmonic Series*(*Amer. Math. Monthly***Vol. 78**,*no. 8*: pp. 864 – 870) www.jstor.org/stable/2316476

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $23 \cdotp 103 \, 45 \ldots$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $23 \cdotp 10345 \ldots$