A modification of the harmonic series that causes it to converge. [duplicate]

Solution 1:

There are $(b-2)(b-1)^{k-1}$ elements $n$ which aren't in $S$ and have exactly $k$ digits, and for each of these you have $\frac{1}{n} \le \frac{1}{b^k}$. So you have an absolutely convergent majorant: $$\sum_{n \notin S} \frac{1}{n} \le (b-2)\sum_{k=1}^{\infty} (\frac{b-1}{b})^k < \infty$$

Solution 2:

First, consider numbers $n$ that are in the range $[b^k, b^{k+1})$. Since there are $b-1$ choices for the first digit in the base $b$ representation of $n$ (as zero cannot be leading), $\frac{b-2}{b-1}$ is the ratio of numbers that do not contain a $1$ in the first position. So, these numbers are the only ones still in $(\mathbb{Z}^+-S)$ and therefore the only ones that may still end up in the terms of the series. However, there is still a $\frac{1}{b}$ chance for each of the remaining positions in that number that a $1$ will appear.

Therefore, in the range $[b^k, b^{k+1})$, there are $(b^k-b^{k+1})(\frac{b-2}{b-1})(\frac{b-1}{b})^k$ numbers that will not be in set $S$ and thus have their reciprocal added to the series. Note also that every term in this range must be less than or equal to $\frac{1}{b^k}$. Therefore, the sum of the terms in the range must be less than or equal to the number of terms present in the range times $\frac{1}{b^k}$: $$\sum_{terms\in{[b^k, b^{k+1})}}(terms)^{-1}\leq{(b^k-b^{k+1})\left(\frac{b-2}{b-1}\right)\left(\frac{b-1}{b}\right)^k\left(\frac{1}{b^k}\right)}$$ It follows that: $$\sum_{n\in({\mathbb{Z}^+}-S)}^\infty \frac{1}{n}=\sum_{k=0}^\infty \sum_{terms\in{[b^k, b^{k+1})}}(terms)^{-1} \leq{\sum_{k=0}^\infty\left[ (b^k-b^{k+1})\left(\frac{b-2}{b-1}\right)\left(\frac{b-1}{b}\right)^k\left(\frac{1}{b^k}\right)\right]} $$ The last summation can be simplified to $$\left(\frac{b-2}{b-1}\right)\sum_{k=0}^\infty\left[(b-1)\left(\frac{b-1}{b}\right)^k\right] = (b-2)\sum_{k=0}^\infty\left(\frac{b-1}{b}\right)^k$$. This is a geometric series, which converges to $\frac{b-2}{1-\frac{b-1}{b}} = b(b-2)$. So, we have: $$\sum_{n\in({\mathbb{Z}^+}-S)}^\infty \frac{1}{n}\leq{b(b-2)}<{\infty}$$ We have just seen that the removal of all terms that contains one or more symbol in base $b$ from the harmonic series causes the series to converge in every base. Intuitively, the larger the base, the more options we have. This would mean we have fewer terms removed and a larger sum, which is suggested but not proved by the final inequality. In base 2, there are so few options (none, in fact) that the series has a value of 0 (which is less than or equal to $b(b-2)$).