Hello All,

Two weeks ago we featured a problem submitted by Sandeep S. In our opinion it was such a superlative problem, that everyone of all levels should get to see it.

Here is the problem:

Sandeep's Harmonic SumsFor each positive integer \(n\), let \[H_n = \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}.\] If \[ \sum_{n=4}^{\infty} \frac{1}{nH_nH_{n-1}} = \frac{a}{b} \] for relatively prime positive integers \(a\) and \(b\), find \(a+b\).

This problem is posed by Sandeep S.

The SolutionNote that \[\begin{align} \sum_{n=4}^{\infty}\frac{1}{nH_nH_{n-1}} &=\sum_{n=4}^{\infty}\frac{1/n}{H_nH_{n-1}} \\ &= \sum_{n=4}^{\infty}\frac{H_n - H_{n-1} }{H_nH_{n-1}} \\ &= \sum_{n=4}^{\infty} \left( \frac{1}{H_{n-1}}-\frac{1}{H_n} \right)\\ &= \frac{1}{H_3} \\ &= \frac{6}{11}. \\ \end{align} \]

Therefore the answer is \(6+11=17\).

The elegance of the solution blew us all away. Thanks Sandeep for such a cool problem!

## Comments

Sort by:

TopNewestWell, I am not going to derogate his method but I don't think I am blown away by this, coz I have done it in the exactly same manner. This was quite an easy problem for 180 points. – Christopher Johnboy · 3 years, 4 months ago

Log in to reply

– Sebastian Garrido · 3 years, 4 months ago

The mind blowing aspect is not the extreme creativity of the solution(Like other commenters pointed out, it is not so creative). Is the fact that intimidating problems everywhere in life can have simple solutions.Log in to reply

A note, this was from the February NIMO contest, featuring only student-written problems. Check it out: internetolympiad.com – Michael Tang · 3 years, 4 months ago

Log in to reply

– Sotiri Komissopoulos · 3 years, 4 months ago

Yeah, the problem was part of the February 2013 NIMO (http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=194&year=2013). It's a cool problem. The original was posed by a user named ssilwa. Is that Sandeep?Log in to reply

– Arshdeep Duggal · 3 years, 4 months ago

Sandeep is an indian name and we call our maths olympiad as INMO.So he is clean...Log in to reply

– Fiyi Adebekun · 3 years, 4 months ago

I know ssilwa from AoPSLog in to reply

– Pranav Arora · 3 years, 4 months ago

Thanks for that link, can you provide a link with the answers too?Log in to reply

– Nur Muhammad Shafiullah · 3 years, 4 months ago

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2944235#p2944235Log in to reply

– Pranav Arora · 3 years, 4 months ago

I meant the solution to the other problems given in the link posted by Sotiri K.Log in to reply

– Nur Muhammad Shafiullah · 3 years, 4 months ago

You can click on the problem number of any problem on the link given by him to see the full solution.Log in to reply

– Pranav Arora · 3 years, 4 months ago

Thanks!Log in to reply

– Oikj P · 3 years, 4 months ago

thanksLog in to reply

Really , it was quite a simple problem and required nothing so creative . This telescopic series are standard way to solve these problems , i think there are many good problems by Zi song or others. :) – Shivang Jindal · 3 years, 4 months ago

Log in to reply

i like this problem so much, it feels like blowing my mind away. i do beg for more creative problem such as this. thank you brilliant.org – Joshua Richard Theodoroes · 3 years, 4 months ago

Log in to reply

this question is from one of maths olympiad exam – Anubhav Singh · 3 years, 4 months ago

Log in to reply

I think there should be at least one comment about limit process, convergence... – Victor Chaves · 3 years, 4 months ago

Log in to reply

I don't see how we go from line 3 to line 4 in the solution, can someone explain? – Ovi N. · 3 years, 4 months ago

Log in to reply

– Liam Ashby · 3 years, 4 months ago

I struggled with that too but if you start to write out the series in full it is (1/H3 - 1/H4) + (1/H4 - 1/H5) + (1/H5 - 1/H6) ....etc so you can see that after !/H3 all the subsequent terms cancel out....leaving 1/H3Log in to reply

– Michael Tong · 3 years, 4 months ago

To elabroate, it leaves out all except 1/H3 and 1/H(infinity), but since the Harmonic series diverges then 1/H(infinity) is zero, so it's 1/H3 - 0 = 1/H3 = 6/11Log in to reply

It's cool. Amazing – Pebrudal Zanu · 3 years, 4 months ago

Log in to reply

a nice one – Namrata Sak · 3 years, 4 months ago

Log in to reply

yeah very cool – Christian Baldo · 3 years, 4 months ago

Log in to reply

Firstly, this is a standard partial fraction trick. Secondly I sincerely question "is there really a reason to be blown away by an algebraic manipulation even though it turns out to be genuine?" (I don't think so) And again to students submitting problems, as I've also earlier mentioned, please please mention the source if you are 'picking up' problems. Brilliant is now a large community and some one or the other will report it. Recently I reported such a case, and Calvin had to change the wording from "posed by

" to "shared by". – Abhishek De · 3 years, 4 months agoLog in to reply

This solution I think the best – Harry Setiawan · 3 years, 4 months ago

Log in to reply

there is 9 dots in 3*3 form how many ways can we draw lines that passes at least 4 dots? – Mohammad Davoodi · 3 years, 4 months ago

Log in to reply

I think everyone who was not mind blown is jealous of the ingenuity of Sandeep – Fiyi Adebekun · 3 years, 4 months ago

Log in to reply

It's one of those problems that if you know to rearrange parts and do a method like the one in the solution then it's easy to get, if not then it is basically impossible to do and when you look at the solution the only thing you learn is "well, I'll do something like that next time I encounter a problem like this." Not too amazing, really. Amazing problems are the ones which you can do in multiple ways depending on how you look at it, ones where you find the solution from small pieces of information you gather over examining pieces of the problem. – Michael Tong · 3 years, 4 months ago

Log in to reply

I felt it quite easy – Shourya Pandey · 3 years, 4 months ago

Log in to reply

– Vamsi Krishna Appili · 3 years, 4 months ago

but ., we go it wrong!Log in to reply

– Superman Son · 3 years, 4 months ago

yepLog in to reply

Really awesome ! – Ritvik Choudhary · 3 years, 4 months ago

Log in to reply

Great technique bro! – Piyal De · 3 years, 4 months ago

Log in to reply

but very easy prob ...i solved in 5 mins – R Kumar · 3 years, 4 months ago

Log in to reply

Uh,to be honest, i tried it the same way.This is a stanadard approach taught in sequence and series problems.So i am not blown away. – Arshdeep Duggal · 3 years, 4 months ago

Log in to reply

– Rafael Saboya · 3 years, 4 months ago

Telescopic sums to solve series really are a standard approach,the problem is to see it as one of them.Log in to reply

– Pranav Arora · 3 years, 4 months ago

Yes, I myself did it the same way so I cannot see what is "mind blowing" here.Log in to reply

– Krishna Jha · 3 years, 4 months ago

http://www.youtube.com/watch?v=eqjl-qRy71w.... this is mind blowing(not related to this problem)....Log in to reply

Duh,I get the same answer using python 2.7 interpreter, i just combine summation,oo and symbols class in sympy module.:) – Mharfe Micaroz · 3 years, 4 months ago

Log in to reply