Math Is Fun Forum

alice 9

Member

Registered: 2010-10-06

Posts: 6

monotone

xn = 1, 1/2,1/3,.........sum ∑1/n
show that xn is monotone but not converge

let
0<xn<7  fore each n by bolzano theorem B-W
xn has convergent sub sequence in what interval the limit of this sequence lie?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: monotone

Hi alice 9;

For part of your question:

It is well known that the harmonic series diverges.
Once you prove that it is monotone decreasing then you can use Nicole Oresme's proof or the integral test.
Or here is a new interesting proof, I do not remember where I saw it from.

Assume that The harmonic series converges to some finite value called S.

S = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ...

If you group it like this:

S = ( 1 + 1/2 ) + (1/3 + 1/4) + (1/5 + 1/6) + ... It is clearly greater than

T = (1/2 +1/2) + (1/4+1/4)  + (1/6 + 1/6) + ...

So S > T but T = 1 + 1/2 + 1/3 + 1/4 + ... = S

So S > S which is a contradiction.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

alice 9

Member

Registered: 2010-10-06

Posts: 6

Re: monotone

thank you bobbym i really appreciate this
and i'm still waiting for  second question can any one solve it in this forum
and that will be greatly appreciated

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: monotone

Hi Alice;

Glad to help so far! What do you know about the Bolzano theorem?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.