Enormous Harmonic Sum proof (help mehhh)

My friend asked me a question about this (from Russia IMO or something I forgot). The question asked for $n=2013$ . I know the answer by just trying small numbers and then predict. Here's the question to prove:

Prove that $\sum\limits_{j=1}^n (\sum\limits_{i=j}^n \frac{1}{i})^{2} + \sum\limits_{i=1}^n \frac{1}{i} = 2n$

If you're confused with these stuffs, it is...

$(\frac{1}{1} + \frac{1}{2} + ... + \frac{1}{n})^{2} + (\frac{1}{2} + ... + \frac{1}{n})^{2} + ....... + (\frac{1}{n})^{2} + \frac{1}{1} + \frac{1}{2} + ... + \frac{1}{n} = 2n$

#HarmonicSeries #Proofs #Summation

Easy Math Editor

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Comments

Let $H_n = \sum_{i=1}^n \frac1{i}$ (with the convention that $H_0 = 0$ ). My approach was to write the quantity on the left side as $S_n = \sum_{j=1}^n (H_n - H_{j-1})^2 + H_n$ . It is enough to show that $S_{n+1}-S_n = 2$ , since $S_1 = 2$ is the base case for an easy induction.

This difference equals $\sum_{j=1}^{n+1} \left( (H_{n+1}-H_{j-1})^2-(H_n-H_{j-1})^2 \right) + \frac1{n+1}$ , which simplifies by a difference of squares formula to $\sum_{j=1}^{n+1} \frac1{n+1} ( H_{n+1} + H_n - 2 H_{j-1}) + \frac1{n+1}$ , which becomes $H_{n+1} + H_n + \frac1{n+1} - \frac2{n+1} \sum_{j=1}^{n+1} H_{j-1} = 2 H_{n+1} - \frac2{n+1} \sum_{j=1}^{n+1} H_{j-1}$ .

Ah, but now we can use the identity $\sum_{i=0}^n H_i = (n+1)H_n - n$ , which is easy to prove by induction. (I found it on the wikipedia page for "harmonic number.") Plugging this in, we get $2 H_{n+1} - \frac2{n+1} ((n+1) H_n - n) = 2 H_{n+1} - 2 H_n + \frac{2n}{n+1} = \frac2{n+1} + \frac{2n}{n+1} = 2$ as desired.

There is probably a much more elegant way to do the problem (i.e. directly without induction), but this certainly works.

Patrick Corn - 7 years, 4 months ago

I don't quite understand how it is enough to prove that $S_{n+1} - S_{n} = 2$ . I understand the rest but I'm just stuck at the first part. =_=" Sorry about that *cries

Samuraiwarm Tsunayoshi - 7 years, 4 months ago

$S_{n+1} - S_{n} = 2(n+1) - 2n = 2$

Ronald Overwater - 7 years, 4 months ago

@Ronald Overwater – Oh, my bad. =__="

Samuraiwarm Tsunayoshi - 7 years, 4 months ago

Thank you for the proof. ^__^

Samuraiwarm Tsunayoshi - 7 years, 4 months ago

This is from IMC 2013. :)

Zi Song Yeoh - 7 years, 4 months ago

Thanks! ^__^

Samuraiwarm Tsunayoshi - 7 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$