Wikipedia:Reference desk/Archives/Mathematics/2006 November 29

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November 29

Problem obtaining the sum of a Complex Series

I have been trying to sum the geometric series:

Sn= e^ix+e^2ix+e^3ix+...+e^nix

I managed to obtain the expression:

Sn= e^ix(e^nix-1)/(e^ix-1)

But I am unsure how to simply this expression.

Thanks for your help.

138.253.175.97 01:46, 29 November 2006 (UTC)

I am looking for new friends. Hope 03CpFns.pdf would help. Twma 04:23, 29 November 2006 (UTC)

You can use the Euler formula to help get rid of the complex values. 137.99.174.5 05:52, 29 November 2006 (UTC)

Before applying Euler's formula (which is a good idea), multiply denominator and numerator both by e^−ix−1 and "multiply" both out (using distributivity of multiplication over addition) so as to write each as a sum of powers of e^ix. That helps to bring the result into the form U + iV and keep it simple without needing to apply trigonometric identities.  --Lambiam^Talk 06:24, 29 November 2006 (UTC)

Conjecture about generalized eigenvectors

If v is a generalized eigenvector of a matrix M, with generalized eigenvalue λ, does that imply that

$\lim _{n\rightarrow \infty }{\frac {|M^{n+1}\mathbf {v} |}{|M^{n}\mathbf {v} |}}=\lambda ?$ —Keenan Pepper 05:43, 29 November 2006 (UTC)

I think that's correct, assuming that you mean

$\lim _{n\rightarrow \infty }{\frac {|M^{n+1}\mathbf {v} |}{|M^{n}\mathbf {v} |}}=|\lambda |$

and you exclude v = 0 and λ = 0. I'd prove it by first estabilishing the formula when M is a Jordan block (split the Jordan block in a diagonal and a nilpotent part) and then using Jordan decomposition for the general case. Disclaimer: I didn't actually prove it myself; it just seems correct to me.-- Jitse Niesen (talk) 07:25, 29 November 2006 (UTC)

mathematics (proof that n^2 is divisible by 3)

prove that 3 divides n is a necessary and sufficient condition for 3 divdes n^2

I assume that n is an integer.

Let's say n/3 = i (integer). Then, n = 3i. In that case, what does n^2 equal ?

That proves it is sufficient. To prove it is necessary, let's say n/3 = r (real, non-integer). Then n = 3r. In this case, what does n^2 equal ? And is that a real number or an integer ? StuRat 13:20, 29 November 2006 (UTC)

To prove the necessary part, you have to show that if n^2 is a multiple of 3 then n must also be a multiple of 3. This is a bit harder than the sufficient part, because it is not true for every divisor. For example, 4 is a factor of 6^2=36, but 4 is not a factor of 6 - so 4 can be a factor of n^2 without being a factor of n. So ... what is the important difference between 3 and 4, and how can you use this difference in your proof of the necessary part ? Gandalf61 13:28, 29 November 2006 (UTC)

A natural tool to use here is the fundamental theorem of arithmetic, which tells us that every integer greater than 1 has a unique factorization as a product of prime powers. Compare the factors of n with those of n². --KSmrq^T 14:09, 29 November 2006 (UTC)

For a more elementary proof of the necessary part, look at the remainder, after division by 3, of n², for the values of n not divisible by 3. What do you observe? Prove that this is not just the case for the few values you tried, but that it always holds. To understand more about why, read the article on modular arithmetic.  --Lambiam^Talk 21:41, 29 November 2006 (UTC)

Hey, did y'all realize that the square root of three is irrational? Proofs for 2] often work for any prime number. PS that's a hint MPS 08:32, 30 November 2006 (UTC)

Assume n/3 doesn't belong to the integers but n²/3 does.

We can write n as the products of its prime factors, P_aP_bP_cP_dP_e...P_z and given that n/3 doesn't belong to the integers and that 3 is prime, P_x =/= 3

Likewise, this means n² = P_a²P_b²P_c²P_d²P_e²...P_z²

Given that P_x =/= 3, and that all P are integers so P_x =/= sqrt(3), then this means that P_x² =/= 3 which contradicts the initial assumption.

Therefore that 3 divides n is a necessary condition that 3 divides n². Likewise, given that n²/3 can be written as n(n/3) and that n is an integer, it is sufficient that n/3 is an integer for n²/3 to be an integer, as the product of two integers is an integer.

Help with force problem

I've been asked to help with some classwork, but just aren't sure. I've outlined the problem below, with the numbers changed so it's not just as given. Feel free to change them again, as it's the principle I want.

A rigid beam weighs 10 N per metre of length. It is horizontal, hinged at one end and lies on a moveable rigid support at the other. A load of 20 N is applied 2 m from the hinged end. What must the length of the beam be to minimise the force applied to the support? 81.153.218.136 15:46, 29 November 2006 (UTC)

Start by drawing a diagram of the forces on the beam. You have:

• The weight of the beam, acting vertically downwards through its centre of mass.
• The 20N load, acting vertically downwards 2m from the hinge.
• The reaction force at the support, acting vertically upwards at the support, which we assume is at the far end of the beam.
• The reaction force at the hinge, which we also know acts vertically upwards (how ?) at the hinge.

Now take moments about the hinge - this allows us to ignore the reaction force at the hinge from here onwards. The beam is in equilibrium, so the moment about the hinge must be 0. The only unknowns in the moment equation are the length of the beam (call this L) and the reaction force at the support (call this F). So you can re-arrange the moment equation to find an expression for F in terms of L. Then you just have to decide what value of L will minimise F (bearing in mind that there is a physical constraint that L is at least 2m).

Conceptually, you can think of the problem as follows. Imagine what would happen if the beam was very light, so you could ignore its weight. Then you could make the reaction force at the support at the far end as small as you liked by making the beam longer and longer. On the other hand, for a very heavy beam (much heavier than the applied load) you would want to make the beam as short as possible. In the problem as given, where the weight of the beam and the applied load are the same order of magnitude, you are between these two extremes. As you lengthen the beam you reach a point at which the increasing weight of the beam "outweighs" the advantage of moving the support away from the hinge. This is why there is a length of beam which minimises the reaction at the support. Gandalf61 17:09, 29 November 2006 (UTC)

Using the data given here, two configurations can be used to compute a check on the reckoning.

1. The beam is 4 m long, so the load is centered and all the downward force (60 N) is evenly split between the end supports.
2. The beam is 2 m long, so the 20 N load force is applied directly at the rigid support, while the 20 N force from the weight of the beam is still split evenly.

It is good practice to look for checks like this. Notice that in both these cases we can compute the downward force at the rigid support without using a formula involving the moment arm. If we get the formula wrong, this will help us catch the error.

These examples might also help us analyze the problem. They suggest the support force opposes two components, one of which acts as in a lever.

Note: If the load force is 40 N, then the support force differs in the two cases, so don't be fooled by a coincidence. --KSmrq^T 00:36, 30 November 2006 (UTC)

Ah, I never thought of taking moments, thus proving how long it is since I knew about this kind of thing. All is now clear, thanks.

Infinite Irrationals Proof

What's the best proof that there are an infinite number of irrational numbers between any two rational values, a, b. THanks, --86.139.127.29 17:02, 29 November 2006 (UTC)

Step 1: prove that we can find a positive irrational number that is smaller than any give positive rational number (so we can find irrational numbers that are "as small as we like"). Step 2: prove that the sum of a rational number (such as a) and an irrational number cannot be rational, so it must be irrational. Gandalf61 17:17, 29 November 2006 (UTC)

Alternate proof: between any two rationals there is another rational (e.g. a + (b-a)/2), so there are an infinite number of rationals between any two, and between any pair of those there is an irrational (e.g. a + (b-a)/π), so there must be an infinite number of those too. —Keenan Pepper 19:38, 29 November 2006 (UTC)

Hmm... for that matter, the numbers a + (b-a)/nπ for positive integers n are already a direct demonstration. It leads to a mental image different from the one I prefer for this question, but it works as advertised. Melchoir 22:46, 29 November 2006 (UTC)

The correct answer is, there is no such thing. "Best" can mean many different things: fewest dependencies, fewest steps, most easily generalized, deepest insight, most inspiring.

To illustrate one alternative, let a and b be two distinct rational numbers, and let [a₀;a₁,…a_m] and [b₀;b₁,…b_n] be their (regular) continued fractions. For example, let

${\begin{aligned}a&{}={\frac {27}{32}}\\&{}=0.84375\\&{}=[0;1,5,2,2],\\b&{}={\frac {73}{87}}\\&{}=0.83908\dots \\&{}=[0;1,5,4,1,2].\end{aligned}}$

We like continued fractions here for several reasons. A real number is rational if and only if its continued fraction terminates. Contrast this with the decimal expansion of b, which does not terminate. Also, if we interpolate between two continued fractions we interpolate between the corresponding reals. For example, let

${\begin{aligned}c&{}={\frac {16}{19}}\\&{}=0.8421\dots \\&{}=[0;1,5,3];\end{aligned}}$

then c lies between our example a and b.

If we append to the continued fraction for c any terminating sequence of positive integers, we get a rational number between a and b. If we append any non-terminating sequence, we get an irrational number. For example,

${\begin{aligned}c'&{}={\frac {53-{\sqrt {3}}}{61}}\\&{}=0.840458\dots \\&{}=[0;1,5,3,{\overline {1,2}}],\end{aligned}}$

where the overline indicates that the terms 1 and 2 repeat, is an irrational number between a and b. Either way, we can easily generate as many of each as we like. (One simple method is to use [0;1,5,3,n] for rationals, and [0;1,5,3,n,1,2] for irrationals.)

Which is "better", this approach or that proposed by Keenan Pepper? Either way we rely on some kind of interpolation. Either way we need to be able to generate as many distinct interpolated values as we like. Either way we need a guarantee that (enough of) the numbers we produce are irrational. Either way we require various pieces of technical machinery, various assumptions.

Should we interpolate decimal fractions instead? The ancient Greeks were comfortable with continued fractions, but they knew nothing of decimal fractions, infinite or not. The situation is reversed for today's students, for whom decimal fractions are familar, with continued fractions unknown. Sometimes what's "best" depends on the audience.

We might also cast the problem in geometric form, where the numbers become lengths or the slopes of lines. This would appeal to our intuition in convincing us that we can interpolate all the values we need, but requires a little care in ensuring irrational values.

Finally, in higher mathematics we discover that the infinity of irrationals dwarfs the infinity of rationals. For example, if we pick a random real value uniformly distributed between a and b, the probability of getting an irrational number is 1, while the probability of getting a rational is 0. This is an enormously important fact, and none of our proofs so far bring it to our attention. The "best" proof probably should. --KSmrq^T 08:37, 2 December 2006 (UTC)

Fun ways to write 1

I'm thinking of a banner to wave at a mathy event. It should say "We're #1", but in some whimsical way, like

WE'RE # $\sum _{n=1}^{\infty }{\frac {6}{\pi ^{2}n^{2}}}$!!

That'd be the analysts' banner, perhaps. What are some other good (not in the sense of "efficient") ways to get to '1'? How would a topologist do it? -GTBacchus^(talk) 20:01, 29 November 2006 (UTC)

(I guess this isn't a reference desk question, in the traditional sense. Oops.) -GTBacchus^(talk) 20:02, 29 November 2006 (UTC)

I'm fond of the series of reciprocals of Sylvester's sequence:

WE'RE # ${\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{43}}+{\frac {1}{1807}}+{\frac {1}{3263443}}+\cdots$

—David Eppstein 20:09, 29 November 2006 (UTC)

A good Wiki one based on the article's feed back is "We're # 0.999...!"

Or how about "Those guys are 0's, but we're # $0^{0}$!"

A computer scientist might say "We're # 1 - It's True!"

And there's always "We're # $-e^{i\pi }$ Dugwiki 21:53, 29 November 2006 (UTC)

Not very fancy but there's also: We're # $i^{2}$. Vespine 21:56, 29 November 2006 (UTC)

Duh, i² is −1. --KSmrq^T 01:18, 30 November 2006 (UTC)

On a tangential corporate teamwork slogan slant: "Real Analyists Unite: There is no $i$ in $\mathbb {R}$." Dugwiki 22:25, 29 November 2006 (UTC)

What about "We're #$0!$"? – b_jonas 23:03, 29 November 2006 (UTC)

I like this one! It even looks like "we're even better than #1, we're #0!", and it makes complete sense to the mathematical inclined. ☢ Ҡi∊ff⌇↯ 23:15, 29 November 2006 (UTC)

If we let the topology be algebraic topology, then the Euler characteristic of the real projective plane, which we can write as

$\chi (\mathbb {P} ),\,\!$

is 1. More cross-specialty is the genus of an elliptic curve, but the notation would be more challenging. Or we have Betti numbers.

If we prefer set theory, we can use cardinality, as in 2^|∅|.

Of course, the natural numbers commonly start with 0, not 1, lending additional support to my final comment:

I think “We’re number 0!” looks like a winner. --KSmrq^T 01:16, 30 November 2006 (UTC)

Here's another cute one: $\displaystyle 1=2\sinh \ln \varphi$. —David Eppstein 03:54, 1 December 2006 (UTC)

$1=-e^{i\pi }=\lim _{n\rightarrow \infty }{\sqrt[{n}]{n}}=\lim _{x\rightarrow 0}{\frac {\sin x}{x}}=\sum _{n=1}^{\infty }2^{-n}$=$\int _{0}^{\pi /2}\sin xdx=\int _{0}^{\infty }e^{-x}dx$ --84.189.249.27 21:23, 1 December 2006 (UTC)

Fibonacci sums

Is there a neat way of finding the sum to infinity of the reciprocals of the Fibonacci numbers?

According to Fibonacci number#Reciprocal sums, no. You can find references at http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html. (updated 25/12/2008 with a closed form using q-Polygamma functions). Melchoir 22:16, 29 November 2006 (UTC).

Actually, let me tweak my answer; it's not so much "no" as "not yet". Melchoir 22:17, 29 November 2006 (UTC)

Melchoir's answer — and our article — may be wrong; see my additional post below for a possible formula.

In any event, there are efficient ways to compute a numeric result, without requiring a closed form.

Here's one: Sum a handful of the first terms, say a dozen. The tail is very nearly a geometric series in the golden ratio, Φ =1.618…, since

$F(n)={\frac {\Phi ^{n}-(1-\Phi )^{n}}{\sqrt {5}}},$

and the (1−Φ)ⁿ component rapidly becomes negligible. (In fact, F(n) is always ^Φn⁄_√5 rounded to the nearest integer.) So approximate the tail as

$\sum _{k=n+1}^{\infty }\left({\frac {\sqrt {5}}{\Phi }}\right)^{k}={\sqrt {5}}\,\Phi ^{1-n}.$

For example, the first 12 Fibonacci numbers are 1, 1, 3, 5, 8, 13, 21, 34, 55, 89, 144; the sum of their reciprocals is ^3651532639⁄_1090449360, approximately 3.34864944026; the geometric tail sums to √5 Φ⁻¹¹, approximately 0.01123623877; and these combine to give approximately 3.35988567903, which agrees with the correct 3.35988566624… to eight decimal places.

We note that 1−Φ is −¹⁄_Φ, so the Fibonacci numbers alternate above and below the geometric terms; this helps with convergence and error bounds. When summing the reciprocals, be sure to start at the small end, here ¹⁄₁₄₄; floating point arithmetic is not associative, and this order will give more accuracy. --KSmrq^T 03:22, 30 November 2006 (UTC)

Almost: there are neat ways to write the sum of certain modifications of the Fibonacci sequence. For example, using the Jacobi theta functions, we can write the sum of every odd-indexed Fibonacci number as

$\sum _{k=0}^{\infty }{\frac {1}{F_{2k+1}}}={\frac {5}{4}}\vartheta _{2}^{2}\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)$

and the sum of squared reciprocal Fibonacci numbers as

$\sum _{k=1}^{\infty }{\frac {1}{F_{k}^{2}}}={\frac {5}{24}}\left(\vartheta _{2}^{4}\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)-\vartheta _{4}^{4}\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)+1\right).$

If we cheat by adding a "1" to each Fibonacci number in the first sum, there is also the closed form

$\sum _{k=0}^{\infty }{\frac {1}{F_{2k+1}+1}}={\frac {\sqrt {5}}{2}},$

and there is a nice nested sum of squared Fibonacci numbers,

$\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{\sum _{j=1}^{k}{F_{j}}^{2}}}={\frac {{\sqrt {5}}-1}{2}}.$

Results such as these make it seem plausible that a neat closed formula for the plain sum of reciprocal Fibonacci numbers could be found, but no one has gotten there yet. Fredrik Johansson 11:03, 30 November 2006 (UTC)

I've copied these over to our article, which presently has no such details; but it would be much appreciated if you can add appropriate references. Also, your reciprocal odd sum has a small error; it should be

$\sum _{k=0}^{\infty }{\frac {1}{F_{2k+1}}}={\frac {\sqrt {5}}{4}}\vartheta _{2}^{2}\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)$

I do not currently have access to a copy, but I believe a relevant reference is

• Borwein, Jonathan M. (July 1998). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley. ISBN 978-0-471-31515-5. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

Apparently it contains a formula due to Landau,

$\sum _{k=1}^{\infty }{\frac {1}{F_{k}}}={\sqrt {5}}\left({\frac {1}{4}}\vartheta _{2}^{2}\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)+L\left({\frac {3-{\sqrt {5}}}{2}}\right)-L\left({\frac {7-3{\sqrt {5}}}{2}}\right)\right).$

I found this in an old online comment by a young Brad Rodgers; however, I'm not sure what L refers to here. Anyone? --KSmrq^T 14:26, 30 November 2006 (UTC)

Yes, that is the right reference. It actually contains much more on reciprocal Fibonacci and Lucas number sums than the formulas listed above (maybe something for a whole Wikipedia article?). I can't find that particular formula in the book (nice find!), but there are a few similar ones involving "L": L denotes a Lambert series with coefficients a_k = 1. By the way, a bunch of additional series are given in the MathWorld article. Fredrik Johansson 17:25, 30 November 2006 (UTC)

$L(\beta )=\sum _{k=1}^{\infty }{\frac {\beta ^{k}}{1-\beta ^{k}}}$

Specifically, this is pages 91–101 of Pi and the AGM. It credits some of them to Landau, E. (1899). "Sur la Série des Invers de Nombres de Fibonacci". Bull. Soc. Math. France. 27: 298–300. —David Eppstein 18:22, 30 November 2006 (UTC)

Quadratic funtions and inequalities

Two trains are traveling in a straight line. The position of the trains as a function of time is given below:

x₁(t)=t²-5t+9

x₂(t)= -t²+t+10

If t>or equal 0 seconds, at what time will the two trains meet?

Some one can help me on this?

Thanks

The trains meet at a time t when their positions are equal; in other words, when x1(t) = x2(t). Do you know what to do with that? Melchoir 22:51, 29 November 2006 (UTC)

I wonder, what sort of bizarre trains are we talking about here? ☢ Ҡi∊ff⌇↯ 23:24, 29 November 2006 (UTC)

Since the trains are hypothetical, I'd say they're "trains of thought". :) Dugwiki 23:38, 29 November 2006 (UTC)

If you get derailed, let us know where you got stuck, and we will help you get back on track. StuRat 10:37, 30 November 2006 (UTC)

You didn't just say that; tell me you didn't! You'll be punished, you know. --KSmrq^T 11:25, 30 November 2006 (UTC)

The inequality qualifier is just stating that the time for the trains can't be negative (i.e. before they actually leave the station). Just set the two equations equal to one another and solve for t. Justin Custer 16:37, 30 November 2006 (UTC)

Finite differences in cylindrical coordinates

Hey guys. I am trying to set up a finite differences calculation in cylindrical coordinates based on the three-dimensional heat equation (I'd write it but I don't know TeX)... but the equation involves a second derivative with respect to r, the radius. How could I find the second derivative with respect to r at the center of the cylinder (when r = 0)? Or does such a thing even exist? I'm not sure if it would exist at, say, the edge of a line, but then again if it's not asymptotic or anything it maybe could be extrapolated from previous values... is this the only way to determine it? Thank you all for your help. Mattb112885 23:20, 29 November 2006 (UTC)

Maybe it is time to learn some TeX markup for formulas. I often cheat by starting from something that looks like what I need, so you could perhaps have copied and edited the equation at the article Heat equation.

You might (hope to) define the second (partial) derivative w.r.t. r at r = 0 by taking one-sided limits, but in the general case you would need to add continuity constraints because the Cartesian-to-polar coordinate transformation is not continuous at r = 0. Are there symmetries you can use, such as that u(r, θ, z) does not depend on θ? If not, maybe cylindrical coordinates are not a good idea. Otherwise, you can use u(r, z) = u(−r, z). Finally, it sometimes helps to use slightly modified coordinates, such as (s, θ, z) where s = r². This may perhaps remove singularities, although the problem of loss of continuity remains in case your heat function is not rotation-insensitive.  --Lambiam^Talk 06:55, 30 November 2006 (UTC)

Yes, the temperature in my problem does not depend on Θ, I'm not sure yet if the dependence on z is important. If I neglect it the formula is (I like your trick, thanks for mentioning it and for pointing me to the page, I hadn't seen that before) ${\partial T \over \partial t}=\alpha \left({1 \over r}{\partial T \over \partial r}+{\partial ^{2}T \over \partial r^{2}}\right)\quad$ Now that I think about it, it would probably be a problem to use this equation at r = 0 anyways, due to division by zero, I'm assuming this is why the mapping is discontinuous? (I always assumed that at r = 0 just corresponded to (x,y) = 0 but never really considered the derivatives before)... is it valid to switch to rectangular just for the purposes of evaluating the second derivative at the center (and some other point near the center) and then using cylindrical for other radii? Thank you for your help. Mattb112885 17:08, 30 November 2006 (UTC)

For this case a switch of variable from r to s = ¹/₂r² really helps. Then ${\partial \over \partial r}=r{\partial \over \partial s}$, so the "division by zero" is avoided.  --Lambiam^Talk 21:08, 30 November 2006 (UTC)

Alright, that will work out well, thank you very much! 137.99.14.70 22:10, 30 November 2006 (UTC)

Since you have ${\partial T \over \partial \theta }=0$, you also have ${\partial ^{2}T \over \partial \theta \partial r}=0$, or ${\partial \over \partial \theta }{\partial T \over \partial r}=0$. Now suppose that $\left.{\partial T \over \partial r}\right|_{r=0}=\alpha \not =0$ (even in a one-sided sense). What would $T$ then look like at the origin? To remove that discontinuity, we must have $\lim _{r\rightarrow 0^{+}}{\partial T \over \partial r}=0$. This might obviate the need for the change of variables to $r^{2}$. Does that help? --Tardis 16:36, 1 December 2006 (UTC)

Well, as remarked above, rotational symmetry implies that $T(r)=T(-r)$, so under the usual assumption that we have a tame function, that derivative indeed vanishes at $r=0$. But how will things behave numerically? A rationale for the change of variables is further that equi-sized cells then have equal area and therefore equal heat capacity. I have no experience with the heat equation for this case, but have observed for similar cases with hyperbolic p.d.e.s that analogous changes of variables increased numerical efficiency and stability.  --Lambiam^Talk 21:15, 1 December 2006 (UTC)