Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

## #26 2007-04-12 17:09:07

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

To 10 :

I think  it converge to

Last edited by Stanley_Marsh (2007-04-12 17:22:40)

Numbers are the essence of the Universe

## #27 2007-04-12 18:40:20

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

The an have to be rational numbers. Are they?

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #28 2007-04-13 00:44:16

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

Awwwww , Let me come up with another one.

Numbers are the essence of the Universe

## #29 2007-04-13 02:07:16

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

You’d be surprised – it’s actually much simpler than you think.

And don’t be afraid to try #9 and #11 – they really aren’t as hard as they look. I reckon #9 can be done in 4 or 5 lines of proof, and #11 in 3 or 4 lines of proof.

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #30 2007-04-13 07:16:53

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

I happened to see the Riemann zeta-function today~lol

Last edited by Stanley_Marsh (2007-04-13 07:17:10)

Numbers are the essence of the Universe

## #31 2007-04-13 07:18:57

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

9 ,11 are hard for me , I haven't that much knowledge of math , I just learn math randomly by myself .haha

Numbers are the essence of the Universe

## #32 2007-04-13 07:51:34

kylekatarn
Power Member

Offline

### Re: Jane’s exercises

#### Zhylliolom wrote:

but a real decent calculator would notify you something like "Warning: 0^0 replaced by 1" .

... like a ti92+ or voyage 200 ; )

## #33 2007-04-13 08:27:24

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

#### Stanley_Marsh wrote:

I happened to see the Riemann zeta-function today~lol

Yes, that’s fine. In fact, you’ve found a required example.

A simpler one would be

The sequence is bounded above (e.g. 2 is an upper bound), it is increasing, and the an are rational – but

is not rational.

Last edited by JaneFairfax (2007-04-13 14:12:50)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #34 2007-04-13 08:42:54

Zhylliolom
Real Member

Offline

### Re: Jane’s exercises

My TI-89 does it... I know a guy with a 92, the thing is huge.

9. Let X = {xn} be a bounded monotone sequence. Since X is bounded, there exists some real M such that xn ≤ M for all n. By the completeness of R, A = sup{xn: n ∈ N} exists and is real. Given ε > 0, A - ε is not an upper bound for X, so we have some xk such that A - ε < xk. Since X is an increasing sequence, xk ≤ xn if k ≤ n. Then A - ε < xk ≤ xn ≤ A < A + ε, so |xn - A| < ε, and hence lim X = A = sup{xn: n ∈ N}.

10. This is almost like your other thread, where we said that (Q, d), where d is the Euclidean metric, is not complete. Anyway, define {an} recursively as a1 = 1 and an+1 = an/2 + 1/an. This is monotone increasing but converges to √2.

11. Trivial. Problem 9 shows that every increasing monotone sequence converges. But every convergent sequence is a Cauchy sequence (this is well-known from analysis, I can prove it if you really want though), so the result follows immediately.

## #35 2007-04-13 13:58:27

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

Brilliant!

Yes, #11 was supposed to contain some sort of hidden trick. I put “rational numbers” there to try and catch the unwary ones off guard. The trick is that all rational numbers are real numbers, so instead of treating (an) as just a rational sequence, treat it as a real sequence. Then, though (by #10) it may not converge in

, it certainly will in
, by #9. Convergence implies Cauchy; thus (an) is a Cauchy sequence.

I’m glad you spotted the trick.

Last edited by JaneFairfax (2007-04-13 14:05:41)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #36 2007-04-16 05:58:16

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #37 2007-04-16 06:18:24

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

Last edited by Stanley_Marsh (2007-04-16 06:26:09)

Numbers are the essence of the Universe

## #38 2007-04-16 06:25:44

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

Oh , forget it ,

Numbers are the essence of the Universe

## #39 2007-04-16 06:39:54

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

But that’s not the question.

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #40 2007-04-16 06:50:26

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

What’s more,

is false. Try a = b = 0, k = 1⁄2.

And

You mean the Cauchy–Schwarz inequality?

Last edited by JaneFairfax (2007-12-19 13:51:06)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #41 2007-04-16 10:51:25

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

I think I got it

Last edited by Stanley_Marsh (2007-04-16 10:59:56)

Numbers are the essence of the Universe

## #42 2007-04-16 19:30:39

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

#### Stanley_Marsh wrote:

Sorry, I don’t get you here.

In the second part, you’re also dividing by a, b, and c. Shouldn’t you perhaps also consider separate cases where each of them is not 0?

Last edited by JaneFairfax (2007-04-16 19:31:13)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #43 2007-04-17 07:54:01

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

actually , the first part  ( They are all positive)

The first part can be done by

U can also think that a,b,c are distinct ,

Last edited by Stanley_Marsh (2007-04-17 08:03:07)

Numbers are the essence of the Universe

## #44 2007-04-17 08:04:11

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

Okay for the first part, but I’m still not happy with the division bit in the second part. If you’re gonna divide, you should make sure you’re not dividing by zero.

As a matter of fact, there is a way to do the second part without doing any kind of division. Try it.

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #45 2007-04-17 08:10:58

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

Numbers are the essence of the Universe

## #46 2007-04-17 08:19:34

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

Yes, you’ve got it!!

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #47 2007-04-26 07:00:24

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

#13

Last edited by JaneFairfax (2007-04-26 19:36:08)

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #48 2007-04-26 10:15:42

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

Don't know it will work tho.

If a ,b,c lie in the same straight line , We have a-b=zc , b-c=xa ,c-a=by , Add them together , get xa+yb+zc=0 ,

Numbers are the essence of the Universe

## #49 2007-04-26 20:07:06

JaneFairfax
Legendary Member

Offline

### Re: Jane’s exercises

I forgot to state that the three vectors a,b,c must be distinct (I’ve edited my post and added it now).

Anyway:

(i) You must show that at least one of x, y, z is not 0. I don’t think you did that. Did you?

(ii) m(x+y+z) = 0 does not imply (x+y+z) = 0. What if m = 0?

(iii) You must also consider the case where all three points lie in the same vertical line. The equation y = kx+m does not cover vertical lines.

(iv) And don’t forget also to prove the converse.

Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

## #50 2007-04-28 01:30:06

Stanley_Marsh
Power Member

Offline

### Re: Jane’s exercises

I have to dicuss those situation separately . Hmmm.....

Numbers are the essence of the Universe