Wikipedia:Reference desk/Archives/Mathematics/2010 February 11

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February 11

Name that series

Does the series $\sum _{n=1}^{\infty }{\frac {n}{2^{n}}}=2$ have a name, such as the harmonic series or the power series does?58.147.58.179 (talk) 04:42, 11 February 2010 (UTC)

I think it's a power series; put x=1/2 and the a's to be the natural numbers Money is tight (talk) 08:40, 11 February 2010 (UTC)

So it is. I linked to the power series, but didn't look at that page because I was erroneously equating them with geometric series, when in fact geometric series are just one example of power series. Thanks. 58.147.58.179 (talk) 10:29, 11 February 2010 (UTC)

And the formula will work if you replace the number 2 on either side of the equation with any complex number ω, provided that the modulus |ω| < 1. You can prove it by finding the radius of convergence of the given power series. •• Fly by Night (talk) 17:28, 11 February 2010 (UTC)

You seem to be a bit confused. The power series $\sum _{n=1}^{\infty }nz^{n}$ is the Taylor series of ${\frac {z}{(z-1)^{2}}}$ about 0, and has radius of convergence 1. Thus, for any z with |z|<1, we have $\sum _{n=1}^{\infty }nz^{n}={\frac {z}{(z-1)^{2}}}$. The OP's series is the z=1/2 case of this. Algebraist 17:41, 11 February 2010 (UTC)

You're exactly right. I have your same exact expression saved in an earlier Matlab file. Not sure why I wrote what I did. Quite embarrassing really. Thank you very much for pointing out that error. •• Fly by Night (talk) 01:35, 12 February 2010 (UTC)

linear programming

I've been using linear programming to find solutions for minimizing ghe cost of making several products which is a comm application. I have noticed, however, that I cannot make just half the resulting number of products for half the minimum cost. How is this phenomenon explained? 71.100.8.16 (talk) 11:06, 11 February 2010 (UTC)

Fixed costs? Or perhaps you're using some kind of batch production that is less efficient for smaller batches? Perhaps there are minimum order quantities for your materials? It's hard to say unless you give more information. —Bkell (talk) 15:05, 11 February 2010 (UTC)

It could be Economies of scope, though that doesn't happen only with the linear case. Pallida  Mors 19:04, 11 February 2010 (UTC)

Symmetry groups

Given a group G, is it possible to construct a space for which G is the symmetry group? I thought at first that this was clearly untrue, since symmetry groups typically have functions as members, and so what does it mean to say that, say, Z mod 5 is the symmetry group of a space? But then I remembered that one of the goals of representation theory is to represent groups as matrix groups, so it didn't seem entirely unlikely. Is there a standard method of constructing the space, if true, and if untrue, does it work for some restricted class of groups we might consider?

The thought occurred to me today during a topology lecture, while discussing the construction of Eilenberg-MacLane spaces, and I wasn't sure where to even begin looking for an answer! (The reference desk seemed a good start...) Thanks for the help, Icthyos (talk) 17:33, 11 February 2010 (UTC)

It's sufficient to say that $\mathbb {Z} _{5}$ is isomorphic to the "real" symmetry group of the space; depending on how generally you mean to construe "space", the obvious example for that case is just 5 separate lines (or points, or planes, or...) with the group elements shuttling points cyclically among them. If you admit such examples, I'm sure any group can be used. --Tardis (talk) 18:23, 11 February 2010 (UTC)

Interpreting "space" very broadly, Cayley's theorem is the canonical result here. Algebraist 18:25, 11 February 2010 (UTC)

It is a theorem of Johannes De Groot that every group is the group of homeomorphisms of a compact Hausdorff space. As Algebraist hints, the idea of De Groot's proof is to use Cayley graphs, but then to replace the edges of the graphs by asymmetric subspaces. —David Eppstein (talk) 02:35, 12 February 2010 (UTC)

David Eppstein, are there more topological spaces than groups? If we map each space to its automorphism group, the theorem of Johannes De Groot shows this is a surjection. Pretty weird I'm sure we can define for each group some space and then show this is a surjection too. Money is tight (talk) 08:53, 12 February 2010 (UTC)

What do you mean by "more"? Both topological spaces and groups form a proper class. —David Eppstein (talk) 04:35, 13 February 2010 (UTC)

I know they form proper class, but that only means we can't talk about them in ZFC. But we can use NBG, where proper classes exist in the object language. In the article Axiom_of_limitation_of_size, it's an example of where we can talk about class functions; i.e. the function itself is a proper class of ordered pairs. Money is tight (talk) 06:25, 13 February 2010 (UTC)

That very axiom tells you that all proper classes have the same cardinality. In this specific example, I think that you don't even need the axiom to prove that both classes are equinumerous with V.—Emil J. 13:38, 15 February 2010 (UTC)

Thanks for the replies, this has given me much to ponder. (Woops, wasn't signed in) Icthyos (talk) 15:28, 14 February 2010 (UTC)

Summing a series

I found by experimentation that the sum of the series 1/2! + 2/3! + ... + n/(n+1)! appeared to be 1 - 1/(n+1)!, then proved this by induction. Can the sum be derived in any other way?—86.166.204.98 (talk) 20:14, 11 February 2010 (UTC)

$\sum _{k=1}^{n}{\frac {k}{(k+1)!}}=\sum _{k=1}^{n}\left({\frac {k+1}{(k+1)!}}-{\frac {1}{(k+1)!}}\right)=\sum _{k=1}^{n}\left({\frac {1}{k!}}-{\frac {1}{(k+1)!}}\right)=1-{\frac {1}{(n+1)!}}$. See also telescoping series. -- Meni Rosenfeld (talk) 21:21, 11 February 2010 (UTC)

(OP)Thanks. I saw how to do it soon after posting by taking the term ${\frac {1}{(n+1)!}}$ from RHS to LHS, whereby everything on the left collapsed to 1.—86.160.104.79 (talk) 13:44, 13 February 2010 (UTC)

Variability of taylor expansions in the complex plane under different branch-cuts

Hi all,

I've just completed the first part of the following problem:

Find the first two non-vanishing coefficients in the Taylor expansion about the origin of the following functions, assuming principal branches for (i), (ii) and (iii), making use of (where appropriate) series expansions for log(1+z), etc.

i)${\frac {z}{log(1+z)}}$

ii)${\sqrt {\cos(z)}}-1$

iii)$\log(1+e^{z})$

iv)$e^{e^{z}}$

Now I just figured they can't ask this question unless the functions are analytic about the origin, in which case the derivative should be the same in all directions at the origin, namely that along the real line, so then I just calculated the Taylor series as if they were real functions, getting:

i)$1+{\frac {z}{2}}$

ii)$-{\frac {z^{2}}{4}}-{\frac {z^{4}}{96}}$

iii)$\log(2)+{\frac {z}{2}}$

iv)$(1+z)e$

However, I didn't use any of the standard series expansions for these calculations, did I make things overly complicated for myself? Anyway, my problem is, I fear I've oversimplified things, because the next part of the question asks how each answer would differ if we didn't take the principal branch in each case? I also need to find the range of values for z for which each series converges. What have I done wrong, because I can't see how my answers would change if we took a different branch cut - are the answers even correct? Thanks very much for any explanation - and also, what would be the most sensible approach for finding the range of convergence for each series?

Many thanks in advance! 131.111.185.68 (talk) 20:20, 11 February 2010 (UTC)

The answers are correct. To hint at where the principal branch comes into play, consider the fact that if a non-principal branch was used, the answers to 1 and 2 could be $-{\frac {iz}{2\pi }}+{\frac {z^{2}}{4\pi ^{2}}}$ and $-2+{\frac {z^{2}}{4}}$. -- Meni Rosenfeld (talk) 21:28, 11 February 2010 (UTC)

euromillions lotto rollover

prize fund is £113 million cost of ticket £1.50 odds of winning 76 million to one. Is it worth purchasing the ticket? —Preceding unsigned comment added by 89.241.185.253 (talk) 21:34, 11 February 2010 (UTC)

Yes, think of all those important government projects you'll be funding !

Mathematically probably not. Although it sounds like a good bet, each time a big lottery gets in such a position it becomes very popular, and a lot of people place bets. So you're not betting on getting £113 million, you're betting on getting a fraction of that. And the bigger the pot the more chance it will be split, the more ways it will likely be split, as more people bet. Every once in a while someone will collect a big pot, but more often it will be won by more than one ticket I think.--JohnBlackburne^words_deeds 22:19, 11 February 2010 (UTC)

People win the lottery without purchasing a ticket. Therefore, the odds of winning with purchasing a ticket and without purchasing a ticket are nearly identical. -- kainaw™ 22:23, 11 February 2010 (UTC)

Just checked for a recent example and these siblings just won $100,000 this week without purchasing a lottery ticket. It was given to them as a birthday present. -- kainaw™ 22:31, 11 February 2010 (UTC)

The odds of winning the jackpot aren't useful, since that doesn't take into account the jackpot being split between multiple people or the smaller prizes. You need to know the total prize fund (the number you've quoted is actually the jackpot) and the total number of tickets bought in order to work out the expectation. A single rollover (which I think this is) is very unlikely to have a positive expectation. Multiple rollovers can, in theory. --Tango (talk) 22:52, 11 February 2010 (UTC)

The concept of the Kelly criterion is you can compute the optimal number of tickets to buy based on the size of your current bankroll and the winning probability and expectation. E.g., if the expected value is positive, then it makes sense to buy more than zero tickets, but since the probability of actually winning is small (the positive expectation comes from the size of the payout if you do win), you should not convert all your savings into lottery tickets. For almost everybody, the optimal number of tickets is much less than 1 ticket (i.e. running the numbers shows you should only bet a fraction of a cent rather than £1.50) and since that fractional bet is not possible, it's better to not bet at all. Buying a lottery ticket turns out to be a good investment (per Kelly) only if you are already a billionaire. 66.127.55.192 (talk) 23:19, 11 February 2010 (UTC)

A bit of math. From the news the jackpot was £85m a week ago. No-one won, but the jackpot gone up by £30m. But only 32% of winnings goes to the prize fund, so about £100m was added to all prize funds. And as only 50% of ticket sales go to winnings, about £200m was spent on tickets over the last week.

At £1.50 a ticket that's maybe 130m tickets bought to date. Still a day to go so probably a few tens of millions, if not more, will be sold this draw. The expected number of winners with 1 in 76 million is probably more than two, and the chance of there being only one winner is maybe less than 50%. And if there's only one winner they have not 1 in 76 million, but 1 in however many tickets sold chance: maybe 1 in 150 million. With the 50% that's a 1 in 300 million chance of scooping the whole jackpot.

I.e. the chance of winning a share of the jackpot is still 1 in 76 million, it's just very likely that it will be shared. Based on the above back of an envelope calculations it still seems a very poor bet (except for the lottery company).--JohnBlackburne^words_deeds 23:23, 11 February 2010 (UTC)

And don't forget about the taxes on the winnings. Rckrone (talk) 17:06, 12 February 2010 (UTC)

There aren't any. --Tango (talk) 23:43, 12 February 2010 (UTC)

And it turns out there were two winners, matching my rough calculations above (though the chance of it being exactly two was I think less than half). One winner was in the UK and won £56 million: at £1.50 a ticket and odds of 76 million to one a poor bet.--JohnBlackburne^words_deeds 23:49, 12 February 2010 (UTC)

The Kelly Criterion is irrelevant - the expectation for a lottery are almost always negative. The Kelly Criterion tells you how high the expectation needs to be for it to be worth you buying a ticket, but we know that will always be a positive number. The only way to have a positive expectation on a lottery (excluding multiple rollovers) is to be the one running the lottery. --Tango (talk) 13:43, 13 February 2010 (UTC)