Linked Questions

5
votes
2answers
5k views

Prove that $1 + \frac{1}{2} + \frac{1}{3} + … + \frac{1}{n}$ is not an integer [duplicate]

Possible Duplicate: Is there an elementary proof that $∑_{k=1}^n 1/k$ is never an integer? Hello, Prove that $1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$ is not an integer. I tried to ...
1
vote
2answers
408 views

Proof $\frac{1}{2}+\frac{1}{3}+…\frac{1}{n}$ is not an integer for integer $n>1$ [duplicate]

I found a way to prove this using Chebychev's theorem, are there ways to solve it without relying on this?
0
votes
1answer
317 views

Sum $\sum \frac{1}{n}\not \in N$ [duplicate]

If $S_n$ denote sum of $n$ terms of H.P. $\frac{1}{2},\frac{1}{3},\frac{1}{4}$ ..... , Then prove using summation of series that $S_n\not\in N$ $\forall \ n \in N$;
2
votes
1answer
225 views

Show that $a_n=\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}$ would not contain a natural number for all n [duplicate]

Show that the series: $a_n=\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ would not contain natural number for all n Can I prove that using "simple tools"?
-2
votes
1answer
136 views

HELP! Prove that the number $\sum_{k=2}^{n}\frac{1}{k}$ is not an integer [duplicate]

Prove that the number $$\sum_{k=2}^{n}{1\over k}$$ is not an integer.
1
vote
0answers
182 views

Let $n>1$, then prove that $1+ 1/2+1/3+\cdots+1/n$ is not an integer. [duplicate]

Possible Duplicate: Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer? Let $n>1$, then prove that $\;1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots +\dfrac{1}{n}\;$ is ...
0
votes
1answer
173 views

If $n>$1 and $S_n= \frac12 + \frac13 +\cdots + \frac1n$, show $S_n$ is not an integer. [duplicate]

At the back of the book which contains this problem, a hint is given to consider $$S_n\cdot2^{k-1}\cdot3\cdot5\cdot9\cdot\ldots$$ where $2^k \le n < 2^{k+1}$. I don't know how this helps? I know ...
0
votes
1answer
104 views

This number is not natural [duplicate]

Let $k$ be a natural number, such that $k>1$. Show that $1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{k}$ is not a natural number. How I can prove this?
1
vote
0answers
115 views

1 + (1/2) + … + (1/n) ; where n>1 [duplicate]

Possible Duplicate: Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer? Does there exist a proof, not by induction, of the fact that $1 + (1/2) + \cdots + (1/n)...
0
votes
1answer
99 views

Proving $f(\mathbb{N}^*)\cap\mathbb{N}=\{1\}$ [duplicate]

Suppose the application $$f:\mathbb{N}^{*}\rightarrow\mathbb{R}$$ $$n\rightarrow\sum_{k=1}^{n}\frac{1}{k}$$How can I prove that $f(\mathbb{N}^*)\cap\mathbb{N}=\{1\}$?
0
votes
0answers
78 views

Harmonic Series is not an integer [duplicate]

I need to show that $H_n=1+{1\over 2}+{1\over 3}+\dots+{1\over n}$ is not an integer for $n>1$. Here I tried $n!H_n=n!+(1.3.5.\dots n)+(1.2.4.\dots n)+\dots+ (12.3.4.\dots (n-1)$ is an integer, $...
0
votes
1answer
35 views

Is there more than one harmonic number that is natural? [duplicate]

n-th harmonic number is: $H_n=\sum_{k=1}^n\frac1k$ is there some $n\neq1$ for which $H_n$ is a natural number? Or can it be proven that there is no such number?
2
votes
1answer
65 views

Let $X=\{x|x=1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}, n\in \mathbb{N}\}$. Find $X \cap \mathbb{N}$ [duplicate]

I am out of hints here. Its trivial to show $1$ is a solution. How to show it's the only solution? Can someone please give me some hints? Please do not use congruence, limits or derivatives because ...
1
vote
0answers
40 views

$X=\{x\mid x=1+1/2+1/3+1/4+\cdots+1/n\}$ ($n\in\Bbb N$); find $X\cap\Bbb N$ [duplicate]

Let $X=\{x\mid x=1+1/2+1/3+1/4+\cdots+1/n\}$ where $n\in\Bbb N$. Find $X\cap \Bbb N$. What I can figure out is to use a general formula and then apply divisibility theorems to solve this, but I ...
1
vote
0answers
32 views

Prove by induction that this sum is not a natural number. [duplicate]

Prove by induction that $1+\frac{1}{2}+\frac{1}{3} +...+\frac{1}{n}$ for any $1<n$ and $n\in \Bbb{N}$ is not a natural number.

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