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July 24

Prove that (1/2) (3/4) (5/6) ... (99/100) < 1/10. (assignment)

Yes, this one is an assignment, so please don't "help" but rather point out where I'm wrong.

I tried to prove the inequality above by observing that

(1) (1/2) (2/3) (3/4) (4/5) ... (99/100) = 1/100,

which is obvious since all inner terms 2..99 cancel out. Similarly,

(3/4) (4/5) ... (98/99) = 3/99 = 1/33,

Next, I define

A = (3/4) (5/6) ... (97/98),
B = (4/5) (6/7) ... (98/99).

That way, one can see that

(2) 0 < A < B,
(3) A B = 1/33.

A < B can be seen from the fact that the factors which make up A are positive and smaller than those which make up B, one factor at a time.

So,

(4) A < 33^–1/2 < 25^–1/2 = 1/5.

Multiplying by (1/2) (99/100), I get

(5) (1/2) A (99/100) < (1/2) (1/5) (99/100) = 99/1000 < 100/1000 = /10. Result!

Line (4) is the one that worries me most, because it is an ugly way to bound A, and so crude that inequality (5) should not work out any more. Still, it does!

Any ideas where I made a mistake? It may be something dumb like bad arithmetics ~~or a sign error~~. No wait, there are no signs. Puzzled, - ¡Ouch! (^{hurt me} / _{more pain}) 08:29, 24 July 2013 (UTC)[reply]

I think line (4) is fine, but you seem to have taken a round about route. You can use a similar argument but starting from

A = (1/2) (3/4) ... (97/98)
B = (2/3) (4/5) ... (98/99)

and using the fact that 1/10 is the square root of 1/100. Gandalf61 (talk) 09:13, 24 July 2013 (UTC)[reply]

That would be A < square root of 1/99.

Which is, now that I think of it, just fine, because then, A < 0.101, and (99/100) A < 0.09999, which is < 0.1 (duh!)

Thanks. - ¡Ouch! (^{hurt me} / _{more pain}) 09:54, 24 July 2013 (UTC)[reply]

Or, even simpler, continue sequences by one more term:

A = (1/2) (3/4) ... (99/100)
B = (2/3) (4/5) ... (100/101)

so that AB = 1/101, then A < sqrt(1/101) < sqrt(1/100) = 1/10. Gandalf61 (talk) 10:12, 24 July 2013 (UTC)[reply]

ARGH! I should have seen that one.

I was thinking, A = assignment, B = baggage. One term of "excess baggage" can't hurt, obviously... - ¡Ouch! (^{hurt me} / _{more pain}) 11:06, 24 July 2013 (UTC)[reply]

Sine integral and differentiation under the integral sign

A textbook exercise is to show that if $\lambda >0$ then $I(\lambda )=\int _{0}^{\infty }{\frac {{\mbox{sin}}\lambda x}{x}}dx$ is independent of $\lambda$ . (The same clearly applies for $\lambda <0$ , but with $I(-\lambda )=-I(\lambda )$ )

The textbook answer is use the substitution $y=\lambda x$ which gives $I(\lambda )=\int _{0}^{\infty }{\frac {{\mbox{sin}}y}{y}}dy$ .

It seemed to me that an alternative method would be to differentiate and show that $I'(\lambda )=0$ except at $\lambda =0$ . But attempting this gives $I'(\lambda )=\lambda \int _{0}^{\infty }{\frac {{\mbox{cos}}\lambda x}{x}}dx$ which doesn't converge. (My first thought before I checked for convergence was to differentiate again and get $I''(\lambda )=-\lambda ^{2}I(\lambda )$ which didn't seem consistent with $I(\lambda )$ being a non-zero constant; that prompted me to check the validity of the previous integral.) Can someone explain exactly why my attempted method didn't work? ManyQuestionsFewAnswers (talk) 15:36, 24 July 2013 (UTC)[reply]

If you differentiate under the integral, you get

\int cos(\lambda x)dx

(which also diverges). Differentiation under the integral sign is not allowed for improper integrals. There are ways around this involving calculation of distributional derivatives. But then the problem is at least as hard as proving directly that I is independent of lambda. Sławomir Biały (talk) 16:02, 24 July 2013 (UTC)[reply]

Why specifically is it not possible to differentiate with respect to

\lambda

? ManyQuestionsFewAnswers (talk) 17:33, 24 July 2013 (UTC)[reply]

I'm not quite clear what the question is, since it seems to me like I've already answered it. The reason is that the hypotheses of theorems that allow the limit operations of differentiation and integration to be interchanged are not met. Generally such limits cannot be exchanged. The "why" only comes in if you want to know why it's allowed in some cases: such cases are very special. Sławomir Biały (talk) 18:10, 24 July 2013 (UTC)[reply]

In "applied mathematics" textbooks, it has to be said that authors are usually a bit debonair about the exchange of integration and differentiation, although I know restrictions apply. My textbook does go on to consider the derivatives with respect to

u

of

I(u)=\int _{0}^{\infty }e^{-xu}dx

for

u>0

, so I didn't think there was a blanket ban on differentiation under the integral sign for improper integrals. ManyQuestionsFewAnswers (talk) 18:22, 24 July 2013 (UTC)[reply]

Well, no there's no blanket assertion that can be made about improper integrals. But there are convergence theorems for these as well that sometimes allow differentiation under the integral. My point is that if you pick an integral "at random", it's not going to be differentiable under the integral. It's the cases of differentiability under the integral that are "special" (and so it makes sense to ask "why" those are special), but to me it doesn't make sense to ask what makes the general case not special: it's a category by exclusion, and there are many different ways a function can be special (since there are many convergence theorems). Sławomir Biały (talk) 18:58, 24 July 2013 (UTC)[reply]

Integrals that crop up in applied maths or statistics textbooks tend to be "nice" or "well-behaved" (the terminology in the texts doesn't get much more technical than that!) and that is what lulled me into my initial false sense of security. I understand your point that the technique requires case by case justification, although in texts like the one I'm working from the conditions to be met are not stated and even "proofs" in the text ignore the necessity - at best, the reader is sometimes assured that the function is "sufficiently nice" for the technique to be valid, but usually even this is dispensed with! I still think it's a sensical question to ask "why does this integral violate the conditions of the best-known theorem(s) for differentiating under improper integrals?" Your earlier answer of "Differentiation under the integral sign is not allowed for improper integrals" is certainly incorrect if "not" is understood as "never". Differentiation under the integral sign is allowed for some improper integrals; I understand that asking exhaustively "which" is difficult, as there may be many theorems that apply, but presumably there are some well-known conditions and this example obviously violates them (and indeed any more obscure sufficient conditions). The article Differentiation under the integral sign is rather unhelpful, since it does not consider improper integrals at all until some crop up without further comment in the "Examples" section. ManyQuestionsFewAnswers (talk) 01:18, 25 July 2013 (UTC)[reply]

A general theorem that applies for some improper integrals can be formulated using the dominated convergence theorem. This will only work for absolutely integrable integrands (that is, functions that are properly Lebesgue integrable) and whose difference quotients are bounded by an integrable function. The integrand $\sin(\lambda x)/x$ is not absolutely integrable, so no such general theorem holds. I'm sorry that you feel that my original statement was misleading: the classical result is the Leibniz rule, which does not work for improper integrals (although it can be improved as I have sketched here to include certain improper integrals). Hope this helps. Sławomir Biały (talk) 11:59, 25 July 2013 (UTC)[reply]

That's brilliant, very helpful thanks. I am going to post a related question later, there is something a little mystifying in the Differentiation under the integral sign. ManyQuestionsFewAnswers (talk) 21:34, 29 July 2013 (UTC)[reply]