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windsylph

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

  • one year ago
  • one year ago

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

    • one year ago
  2. windsylph
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    H_n =1+ 1/2 + 1/3 + ... 1/n

    • one year ago
  3. windsylph
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    H_n is the actual sum from k=1 to n of 1/k

    • one year ago
  4. Limitless
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    So: \[H_n=\sum_{1 \leq i \leq n}\frac{1}{i}\]

    • one year ago
  5. Limitless
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    Have you read Concrete Mathematics?

    • one year ago
  6. windsylph
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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.

    • one year ago
  7. Limitless
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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.

    • one year ago
  8. windsylph
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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

    • one year ago
  9. windsylph
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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

    • one year ago
  10. Limitless
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    Yes, that is step one. It requires manipulating the sum from there.

    • one year ago
  11. windsylph
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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?

    • one year ago
  12. Limitless
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    Wait, first. Do you know what the Iverson Bracket is?

    • one year ago
  13. windsylph
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    idk if I even typed that right.. aside from indexing errors

    • one year ago
  14. windsylph
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    Sorry, but no

    • one year ago
  15. Limitless
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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: \[ [x]= \left\{ \begin{array}{c} 1 &\text{if } x \text{ is true},\\ 0 &\text{otherwise.} \end{array} \right. \]

    • one year ago
  16. Limitless
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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?

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

    • one year ago
  18. windsylph
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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

    • one year ago
  19. Limitless
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    Ah, yes. Induction is much easier. But this makes sense of _why_ it is how it is.

    • one year ago
  20. Limitless
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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.

    • one year ago
  21. windsylph
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    Haha thanks.. Sorry for the trouble. Maybe I should've mentioned induction earlier

    • one year ago
  22. Limitless
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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.

    • one year ago
  23. windsylph
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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

    • one year ago
  24. Limitless
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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

    • one year ago
  25. windsylph
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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!

    • one year ago
  26. Limitless
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    You're welcome. :D

    • one year ago
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