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windsylph Group Title

Prove that the sum of n harmonic numbers H_1 + H_2 + ... + H_n = (n+1)H_n - n

  • 2 years ago
  • 2 years ago

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  1. yakeyglee Group Title
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    What do you define as a harmonic number? \(H_n= \frac{1}{n}\)?

    • 2 years ago
  2. windsylph Group Title
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    H_n =1+ 1/2 + 1/3 + ... 1/n

    • 2 years ago
  3. windsylph Group Title
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    H_n is the actual sum from k=1 to n of 1/k

    • 2 years ago
  4. Limitless Group Title
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    So: \[H_n=\sum_{1 \leq i \leq n}\frac{1}{i}\]

    • 2 years ago
  5. Limitless Group Title
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    Have you read Concrete Mathematics?

    • 2 years ago
  6. windsylph Group Title
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    No, I'm in a Discrete Mathematics course right now. This is one of the challenge homework problems in the book.

    • 2 years ago
  7. Limitless Group Title
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    Concrete Mathematics has an entire chapter dedicated to summation. You could learn various great concepts. This is even one particular problem. I recommend the book to you. I could attempt to explain the method, if you would like. It is a weebit complex, however.

    • 2 years ago
  8. windsylph Group Title
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    I guess you could try.. the course assumes the student has a math background up to Calculus I. So far I've completed three calculus courses, ordinary differential equations, and linear algebra. Not sure if this problem pulls material from these latter courses though. Even so I doubt I'll understand what you're about to explain lol

    • 2 years ago
  9. windsylph Group Title
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    But basically, a summation of a summation, right? Although I don't think the way to approach that is basic at all haha

    • 2 years ago
  10. Limitless Group Title
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    Yes, that is step one. It requires manipulating the sum from there.

    • 2 years ago
  11. windsylph Group Title
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    \[\sum_{j=1}^{n} H_n = \sum_{j=1}^{n} \sum_{k=1}^{j} \frac{1}{k}\] Lol, what's next?

    • 2 years ago
  12. Limitless Group Title
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    Wait, first. Do you know what the Iverson Bracket is?

    • 2 years ago
  13. windsylph Group Title
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    idk if I even typed that right.. aside from indexing errors

    • 2 years ago
  14. windsylph Group Title
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    Sorry, but no

    • 2 years ago
  15. Limitless Group Title
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    It's okay. Not many people do. But it is a notation used in summation to make things easier. The idea is this: \[ [x]= \left\{ \begin{array}{c} 1 &\text{if } x \text{ is true},\\ 0 &\text{otherwise.} \end{array} \right. \]

    • 2 years ago
  16. Limitless Group Title
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    Okay, so this is relevant because it allows summation to be manipulated easily. When we say, for example, \(\sum_{1 \leq i \leq n}i\), we can equivalently say (using this notation) \[\sum_{i} i [1\leq i \leq n]\] This works because we take \(i\) to be evaluated from \(-\infty\) to \(\infty\) and the Iverson Bracket is zero at all points which do not satisfy our condition. Make sense?

    • 2 years ago
  17. Limitless Group Title
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    (Oh, by the way, \(\sum_{1 \leq i \leq n}i\) is the same as \(\sum_{i=1}^{n}i\).)

    • 2 years ago
  18. windsylph Group Title
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    actually.. we may actually be able to prove it without using that. the chapter I'm working on is about proofs by induction. a simple algebraic manipulation may work

    • 2 years ago
  19. Limitless Group Title
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    Ah, yes. Induction is much easier. But this makes sense of _why_ it is how it is.

    • 2 years ago
  20. Limitless Group Title
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    For induction, check \(n=1\). Then, assume for some integer \(k\), that the statement is true. Show that by adding \(H_{k+1}\) to the left side results in the right side.

    • 2 years ago
  21. windsylph Group Title
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    Haha thanks.. Sorry for the trouble. Maybe I should've mentioned induction earlier

    • 2 years ago
  22. Limitless Group Title
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    No, it's fine. You should learn about Iverson Brackets and etc anyway. Trust me, it makes almost every summation ridiculously easier.

    • 2 years ago
  23. windsylph Group Title
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    Thanks, that might help in later computer science classes. The course I'm taking right now is like discrete math for comp sci

    • 2 years ago
  24. Limitless Group Title
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    Concrete Mathematics is all about Discrete Mathematics. Take a look here on Wiki about it: http://en.wikipedia.org/wiki/Concrete_Mathematics

    • 2 years ago
  25. windsylph Group Title
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    Oh wow, then I might really come across it in my later years of study. Thank you very much for the info!

    • 2 years ago
  26. Limitless Group Title
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    You're welcome. :D

    • 2 years ago
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