Harmonic progression

Suppose there is a 1m1m rubber band with an ant on it. Every time the ant walks 1cm1cm, the rubber band expands and the circumference increases by 1m1m, 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%1\% of the entire rubber band. Then, it went 0.5%0.5\%. Then, 0.333%0.333\%, 0.25%0.25\%, 0.2%0.2\%... Add them up: 1100+12100+13100+\frac{1}{100}+\frac{\frac{1}{2}}{100}+\frac{\frac{1}{3}}{100}+\cdots =1100×(1+12+13+)= \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+12+13++1a=1001+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{a} = 100 01(1+x+x2+x3++xa1) dx=100\int_{0}^{1}{(1+x+x^2+x^3+\cdots+x^{a-1}) \ dx = 100} 01xa1x1 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 - γ\gamma. So ae100a \approx e^{100}.

Note by Raymond Fang
4 months, 1 week ago

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