Associated Topics || Dr. Math Home || Search Dr. Math

### Proof Involving Sums of Reciprocals

```Date: 09/13/2004 at 11:29:14
From: Drew
Subject: Sums of Reciprocals

For n > 1, prove that the sum of reciprocals from 1 to n does not
sum to an integer.  For n = 4, for example, prove that 1/1 + 1/2 +
1/3 + 1/4 does not sum to an integer.

I have tried a few methods of approach.  I have tried the most
obvious, induction.  I found a theorem that says that if gcd(a, b) =
gcd(c, d) = 1, then a/b + c/d is an integer only if the absolute
value of b and d is equal.  So, in the inductive step, I am adding
on 1/(k+1) which fits the c and d above, but I can't think how to
prove that when you reduce the "k" step (from the inductive
hypothesis), that the denominator of that fraction is not equal to
(k+1).  That was my induction work.

I have tried some other things too.  Some numerical analysis to see
if there was some pattern to the fractions, but there seems to be
none.

```

```
Date: 09/13/2004 at 14:18:42
From: Doctor Vogler
Subject: Re: Sums of Reciprocals

Hi Drew,

Thanks for writing to Dr. Math.  I would do this by starting with

n   1
sum --- = M
i=1  i

and then multiplying both sides by n!/k (where k divides n!) so that
everything is an integer, including all of the terms of the sum on the
left *except one*.  Bertrand's Postulate can be helpful here, but you
don't even need that.  You can do it with k being just the right power
of two.

Does that help?

back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search