Question

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

Answer 1

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

Answer 2

H_n =1+ 1/2 + 1/3 + ... 1/n

Answer 3

H_n is the actual sum from k=1 to n of 1/k

Answer 4

So: \[H_n=\sum_{1 \leq i \leq n}\frac{1}{i}\]

Answer 5

Have you read Concrete Mathematics?

Answer 6

No, I'm in a Discrete Mathematics course right now. This is one of the challenge homework problems in the book.

Answer 7

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.

Answer 8

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

Answer 9

But basically, a summation of a summation, right? Although I don't think the way to approach that is basic at all haha

Answer 10

Yes, that is step one. It requires manipulating the sum from there.

Answer 11

\[\sum_{j=1}^{n} H_n = \sum_{j=1}^{n} \sum_{k=1}^{j} \frac{1}{k}\]

Lol, what's next?

Answer 12

Wait, first. Do you know what the Iverson Bracket is?

Answer 13

idk if I even typed that right.. aside from indexing errors

Answer 14

Sorry, but no

Answer 15

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

Answer 16

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?

Answer 17

(Oh, by the way, \(\sum_{1 \leq i \leq n}i\) is the same as \(\sum_{i=1}^{n}i\).)

Answer 18

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

Answer 19

Ah, yes. Induction is much easier. But this makes sense of _why_ it is how it is.

Answer 20

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.

Answer 21

Haha thanks.. Sorry for the trouble. Maybe I should've mentioned induction earlier

Answer 22

No, it's fine. You should learn about Iverson Brackets and etc anyway. Trust me, it makes almost every summation ridiculously easier.

Answer 23

Thanks, that might help in later computer science classes. The course I'm taking right now is like discrete math for comp sci

Answer 24

Concrete Mathematics is all about Discrete Mathematics. Take a look here on Wiki about it: http://en.wikipedia.org/wiki/Concrete_Mathematics

Answer 25

Oh wow, then I might really come across it in my later years of study. Thank you very much for the info!

Answer 26

You're welcome. :D