# Harmonic progression

Suppose there is a $1m$ rubber band with an ant on it. Every time the ant walks $1cm$, the rubber band expands and the circumference increases by $1m$, and the length of the ant from the starting point also becomes longer. So, can the ants complete a circle? I think it can't do it, but it can. First, it takes $1\%$ of the entire rubber band. Then, it went $0.5\%$. Then, $0.333\%$, $0.25\%$, $0.2\%$... Add them up: $\frac{1}{100}+\frac{\frac{1}{2}}{100}+\frac{\frac{1}{3}}{100}+\cdots$ $= \frac{1}{100} \times (\color{#D61F06}{1+\frac{1}{2}+\frac{1}{3}+\cdots})$ It is a harmonic series, and the result is positive infinity. But we don't need positive infinity, we just need to make it greater than 100. Let $1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{a} = 100$ $\int_{0}^{1}{(1+x+x^2+x^3+\cdots+x^{a-1}) \ dx = 100}$ $\int_0^1 {\frac{x^a-1}{x-1} \ dx = 100}$ Then-I don't know how I should find the anti-derivative of this score! If anyone knows, welcome to answer in the comment area! But we also know that Euler-Mascheroni constant - $\gamma$. So $a \approx e^{100}$. Note by Raymond Fang
4 months, 1 week ago

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