$$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots$$
Okay so we all know the harmonic series is divergent right? But apparently when you remove all the terms that has a nine in it, suddenly it becomes a convergent series.
Now this can easily be deduced via a geometric sum seen in https://en.wikipedia.org/wiki/Kempner_series .
$$\sum_{\text{$n$ does not contain $9$}} \frac{1}{n} < 8 \sum_{n = 1}^{\infty} \left(\frac{9}{10}\right)^{n-1} = 80.$$
What I find counter intuitive is the fact that when you literally take only 100 of the terms of the harmonic series (not arbitrarily of course) $$$$\frac 11\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots$$ \frac 1{100}\sum_{n=1}^\infty\frac1n=\frac 1 {100} + \frac 1{200} + \frac 1{300} + \cdots$$
i.e. 1/100 + 1/200 + 1/300 +..... or any number bigger than that, you get a series which is more than or equals to 1/100 multiplied by the harmonic series, which is infinity! So my real question is that is there a special way which the numbers are added or removed which affects the sum of the series? How do you identify/characterize this special form of choosing which numbers to include/exclude to form a convergent/divergent series?