Proving Divergence of Harmonic Series

[Дьявол]

Hello! I got one issue with proving divergence of series. I start covering this part of mathematics and don't understand how to prove it. Here is the issue:
I got one harmonic series:

$$\sum_{n=1}^{\infty}{\frac{1}{n}}=1 + \frac{1}{2} + \frac{1}{3} +...$$
We need to show that the series of partial sums (separate sums) is not bounded.

X_n=1 + 1/2 +1/3 +...+ 1/n

As I can see:

X₂=1 + 1/2 = X₁ + 1/2

but what I can't understand is:

X₄=X_2²=1 + 1/2 + 1/3 + 1/4 > 1 + 1/2 + 2*1/4 = 1 + 2/2

and

X_2^k>1 + k/2 where k>1

Can you please give me a short explanation that would help me understand?

Thanks in advance.

 

[Office_Shredder]

Basically you group up the terms, doubling the number of terms grouped each time. So you start off

(1) - 1 term. Now we add two more

(1) + (1/2 + 1/3) - now we add four more

(1) + (1/2 + 1/3) + (1/4 + 1/5 + 1/6 + 1/7) - now we add 8 more

(1) + (1/2 + 1/3) + (1/4 + 1/5 + 1/6 + 1/7) + (1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15)

note the kth block has as its last term 1/(2^k-1)

Basically, each of those blocks of terms is greater than 1/2. Why?

1 > 1/2 trivially

1/2 + 1/3 > 1/2 trivially too.

1/4 + 1/5 + 1/6 + 1/7... well, 1/4>1/8, 1/5>1/8, 1/6>1/8, 1/7>1/8... so

1/4 + 1/5 + 1/6 + 1/7 > 1/8 + 1/8 + 1/8 + 1/8 =1/2

Similarly for the next block

1/8>1/16, 1/9>1/16... 1/15>1/16 so

(1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15) > 8/16 = 1/2

You can repeat this process indefinitely... the next 16 terms will all be greater than 1/32 ( so adding them up gives something > 16/32), the next 32 terms after that will all be greater than 1/64, so adding them gives something > 32/64, etc.

 

[Дьявол]

Thanks for the post. It is great.

But how do they got this one:

X₄=X_2²=1 + 1/2 + 1/3 + 1/4 > 1 + 1/2 + 2*1/4 = 1 + 2/2
We said that 1 + 1/2 > 1/2, so 1+1/2+1/3+1/4>1/2+1/3+1/4. But it does not match up with the statement above. Can you help me realize it?

Thanks in advance.

 

[mathman]

It is based on the fact that 1/3 > 1/4 and for the next group 1/5, 1/6, 1/7 are all > 1/8, etc.

 

[Werg22]

Alternatively, you can use the integral test. It's easy to see geometrically. 1/n is the area of the rectangle with height 1/n and width 1 on the interval [n, n+1]. Looking at 1, 1/2, 1/3, 1/4, 1/n you can see that each rectangle is over the portion of the region under the curve 1/x delimited by x = n and x = n +1. So the area of the region under 1/x over the interval [1,n+1] is clearly smaller than 1 + 1/2 + ... + 1/n. That area is log (n+1), and that goes to infinity for large n's, so it follows that 1 + 1/2 + ... + 1/n also goes to infinity.

 

[Дьявол]

@Werg22, sorry but I have never learned about integrals. In my next lessons I would probably do it.

@mathman
1+1/2>1/2
and 1/3>1/4
If we sum all of them + 1/4
So 1+1/2+1/3+1/4>1/2+1/4+1/4=1/2+1/2=2/2
Where is the 1 ?

 

[Office_Shredder]

1+1/2>1/2
and 1/3>1/4
If we sum all of them + 1/4
So 1+1/2+1/3+1/4>1/2+1/4+1/4=1/2+1/2=2/2

You tell me where the 1 is. You shouldn't have used the inequalities so bluntly

1=1
1/2=1/2
1/3 + 1/4 > 1/2

Hence 1 + 1/2 + 1/3 + 1/4 > 1 + 1/2 + 1/4 + 1/4 = 1 + 1/2 + 1/2 = 1 + 1 = 2

 

[Дьявол]

Sorry for misunderstanding. Now I think I got it right:

1/3 > 1/4

1 + 1/2 + 1/3>1 + 1/2+ 1/4

1+1/2+1/3+1/4>1 + 1/2 + 1/4 + 1/4

So 2ⁿ>1 + n/2

Thanks for the help.