# Showing harmonic series is divergent

1. Jul 17, 2015

### brycenrg

1. The problem statement, all variables and given/known data

2. Relevant equations
Where do the terms 1/4 come from? Are they ambiguous?

3. The attempt at a solution
Trying to understand the text

2. Jul 17, 2015

### RUber

The 1/4 is a convenient comparison since 1/3 > 1/4.
By comparing the series against something easy to calculate, you can show that the series is growing and not converging.

3. Jul 18, 2015

### brycenrg

I'm not to familiar with proofs. But I can say any number? Why don't I just use 1/100? That's obviously smaller

4. Jul 18, 2015

### brycenrg

By showing that the left side is greater than the right what does that show?

5. Jul 18, 2015

### RUber

You are trying to show that the series does not converge.
In order to do that you are replacing some terms with smaller terms to make a known series that does not converge. Therefore the series in question must not converge because it is bigger than another series that doesn't converge.
The pattern you are supposed to see is that :
$\sum_{n=1}^{2^k} 1/n>{k}$

6. Jul 18, 2015

### RUber

If you wanted to replace the terms with 1/100, you would need a lot more terms to see the pattern, but you could still notice that the first hundred terms would be greater than 1, the next 10000 terms would be greater than 1, so the sum would be more than two, the next million terms would be more than 1, for a total sum greater than 3 and so on. The point is that no matter how small the fraction, you won't run out of numbers needed to show you can add to another whole number.

7. Jul 18, 2015

### HallsofIvy

Staff Emeritus
The easiest way to show that the harmonic series does not converge is to use the "integral test". It is easy to show that $$\int_1^\infty \frac{1}{x} dx$$ does not exist so the harmonic series does not converge.