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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
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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yakeygleeBest ResponseYou've already chosen the best response.0
What do you define as a harmonic number? \(H_n= \frac{1}{n}\)?
 one year ago

windsylphBest ResponseYou've already chosen the best response.1
H_n =1+ 1/2 + 1/3 + ... 1/n
 one year ago

windsylphBest ResponseYou've already chosen the best response.1
H_n is the actual sum from k=1 to n of 1/k
 one year ago

LimitlessBest ResponseYou've already chosen the best response.1
So: \[H_n=\sum_{1 \leq i \leq n}\frac{1}{i}\]
 one year ago

LimitlessBest ResponseYou've already chosen the best response.1
Have you read Concrete Mathematics?
 one year ago

windsylphBest ResponseYou've already chosen the best response.1
No, I'm in a Discrete Mathematics course right now. This is one of the challenge homework problems in the book.
 one year ago

LimitlessBest ResponseYou've already chosen the best response.1
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

windsylphBest ResponseYou've already chosen the best response.1
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

windsylphBest ResponseYou've already chosen the best response.1
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

LimitlessBest ResponseYou've already chosen the best response.1
Yes, that is step one. It requires manipulating the sum from there.
 one year ago

windsylphBest ResponseYou've already chosen the best response.1
\[\sum_{j=1}^{n} H_n = \sum_{j=1}^{n} \sum_{k=1}^{j} \frac{1}{k}\] Lol, what's next?
 one year ago

LimitlessBest ResponseYou've already chosen the best response.1
Wait, first. Do you know what the Iverson Bracket is?
 one year ago

windsylphBest ResponseYou've already chosen the best response.1
idk if I even typed that right.. aside from indexing errors
 one year ago

LimitlessBest ResponseYou've already chosen the best response.1
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

LimitlessBest ResponseYou've already chosen the best response.1
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

LimitlessBest ResponseYou've already chosen the best response.1
(Oh, by the way, \(\sum_{1 \leq i \leq n}i\) is the same as \(\sum_{i=1}^{n}i\).)
 one year ago

windsylphBest ResponseYou've already chosen the best response.1
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

LimitlessBest ResponseYou've already chosen the best response.1
Ah, yes. Induction is much easier. But this makes sense of _why_ it is how it is.
 one year ago

LimitlessBest ResponseYou've already chosen the best response.1
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

windsylphBest ResponseYou've already chosen the best response.1
Haha thanks.. Sorry for the trouble. Maybe I should've mentioned induction earlier
 one year ago

LimitlessBest ResponseYou've already chosen the best response.1
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

windsylphBest ResponseYou've already chosen the best response.1
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

LimitlessBest ResponseYou've already chosen the best response.1
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

windsylphBest ResponseYou've already chosen the best response.1
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
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