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windsylph

  • 2 years ago

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

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

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

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

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

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

  6. windsylph
    • 2 years ago
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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.

  7. Limitless
    • 2 years ago
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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.

  8. windsylph
    • 2 years ago
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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

  9. windsylph
    • 2 years ago
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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

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

  11. windsylph
    • 2 years ago
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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?

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

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

  14. windsylph
    • 2 years ago
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    Sorry, but no

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

  16. Limitless
    • 2 years ago
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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?

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

  18. windsylph
    • 2 years ago
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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

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

  20. Limitless
    • 2 years ago
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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.

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

  22. Limitless
    • 2 years ago
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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.

  23. windsylph
    • 2 years ago
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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

  24. Limitless
    • 2 years ago
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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

  25. windsylph
    • 2 years ago
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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!

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

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