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

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...