Math Is Fun Forum

meo_beo

Member

Registered: 2008-06-01

Posts: 4

Prove this

Hi, this one may seem trivial but I really need help. Prove that:

1/11 + 1/12 + 1/13 + .... + 1/80 < 1.5

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Re: Prove this

Well, I think you get the inequality sign for the question wrong.

Assuming what you type is

Observe that

Therefore,

Last edited by coffeeking (2009-02-17 16:02:16)

meo_beo

Member

Registered: 2008-06-01

Posts: 4

Re: Prove this

How can you say that 1/11 + 1/12+ ....+ 1/40 > 4(1/40) ?

It would be ok to say : 1 + 1/2 + 1/3 + 1/4 + ....+ 1/11 + ....+ 1/40 > 40(1/40). But here the sequence starts at 1/11 so I'm afraid there's something wrong with your reasoning.

Last edited by meo_beo (2009-02-17 17:46:28)

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: Prove this

ok, here's my version of proof

sorry if I made any mistakes

Last edited by Dragonshade (2009-02-17 18:35:02)

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Re: Prove this

Oh my gosh... Think I need more coffee you are right meo_beo, I have made a mistake sorry for the confusion.

Anyway, Dragonshade has a great proof. However if I observed correctly this time round, I think the last line of your proof should be

Since from

there are a total of

terms.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Prove this

There are 68 terms because the term corresponding to 1/45 is 1/45.  However, you can't include 1/45 in the sequence twice.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Prove this

Coffeeking's method works with a bit of a tweak.

Group the first ten terms together, then the next twenty and then the other forty.
This gives that the sum is greater than 10(1/20) + 20(1/40) + 40(1/80) = 1/2+1/2+1/2 = 1.5.

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Prove this

Dragonshade’s method looks to be the best so far – although

is logically the wrong way round. You’re supposed to start with

and work backwards to

.

I was thinking maybe you could say that

This is easily justified graphically.

Hence

which sets an even greater lower bound for the sum.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Prove this

Right then, folks. To prove

you only need to prove that

for all positive real numbers

(the sum follows by adding terms piecewise).

First, we prove this:

Proof:

In particular,

for all

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Prove this

Pretty good, Jane!
I also like how you can use similar reasoning to say that the sum is less than ln(80/10) ≈ 2.07944.

So we know that the answer is somewhere in ( ln(81/11), ln(8) ).
Now I'm wondering if there's some way of showing that the answer is close to the midpoint of the interval, and improving the bounds that way.

---

Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Prove this

Well, you can improve on the accuracy by using simple tricks. For example, leaving out the first term,

If you want even better,

etc.