RE: Re: Re: Tough Limit
- To: mathgroup at smc.vnet.net
- Subject: [mg34331] RE: [mg34286] Re: Re: [mg34235] Tough Limit
- From: "DrBob" <majort at cox-internet.com>
- Date: Wed, 15 May 2002 03:35:24 -0400 (EDT)
- Reply-to: <drbob at bigfoot.com>
- Sender: owner-wri-mathgroup at wolfram.com
Vladimir's original numeric experiment was very good (plotting the sinusoidal nature of the number), and he was right not to limit n to integers without reason. It we do limit n to integers, we could do the same experiment with ListPlot, or we could use something like Table[10^k+Random[Integer,{0,10^k}]] to provide examples. Randomly distributed examples should prevent misleading results such as what happened when all the examples were even. Table[n=10^k+Random[Integer,{0,10^k}];N[Binomial[-1/2,n] *Sqrt[n*Pi]],{k,1,4}] {0.992219, -0.99881, 0.999919, -0.999993} (Notice the accidental alternation of signs in this sample!) This is very SLOW if you let k exceed 3, however: Timing[(n=10^#+Random[Integer,{0,10^#}];N[Binomial[-1/2, n]Sqrt[n*Pi]])&[4]] {3.063 Second,-0.999991} Timing[(n=10^#+Random[Integer,{0,10^#}];N[Binomial[-1/2, n]Sqrt[n*Pi]])&[5]] {1377.95 Second,-0.999999} (I'm not EVEN trying the next one.) No problem though, if you make the argument Real rather than integer: Timing[(n = 10^# + Random[Integer, {0, 10^#}]; N[Binomial[-1/2, n]Sqrt[n*Pi]]) &[5.]] {0. Second, -0.999999} So, in a ListPlot or random sample of Integers, you have to turn the integers into Reals before evaluating. For larger n, it could be faster to use the Stirling or Gosper approximation (maybe). Bobby -----Original Message----- From: Vladimir Bondarenko [mailto:vvb at mail.strace.net] To: mathgroup at smc.vnet.net Subject: [mg34331] [mg34286] Re: Re: [mg34235] Tough Limit Dear James, I like very much the way you use: NUMERIC EXPERIMENT. It is a nice research approach. We can learn a lot of important points from such experimenting. It was a way of Euler, Gauss, Ramanujan, and thousands other gifted mathematicians. But to learn ALL (not just a lot) we need to plan our numeric experiments VERY carefully. (And then, ideally, think hard about our numerics, to eschew the insidious TRAPS) You have found that Table[Sqrt[n*Pi]*Binomial[-1/2, n] /. n -> 10^k, {k, 1., 7.}] {0.987583, 0.998751, 0.999875, 0.999988, 0.999999, 1., 1.} while Table[Sqrt[n*Pi]*Binomial[-1/2, n] /. n -> 10^k + 1, {k, 1., 7.}] {-0.988705, -0.998763, -0.999875, -0.999988, -0.999999, -1., -1.} Nice. Now let's have a look at our substitutions n -> 10^k + 0 and n -> 10^k + 1 Why don't try something like n -> 10^k + a ? OK. Let's learn more about this limit. Table[Sqrt[n Pi]*Binomial[-1/2, n] /. n -> 10^k + 1/2, {k, 1., 7.}] {0, 0, 0, 0, 0, 0, 0} Mysterious... Not a vestige of 1 or -1 ... Table[Sqrt[n Pi]*Binomial[-1/2, n] /. n -> 10^k + 1/3, {k, 1., 7.}] {0.49399, 0.499377, 0.499938, 0.499994, 0.499999, 0.5, 0.5} Table[Sqrt[n Pi]*Binomial[-1/2, n] /. n -> 10^k + 1/4, {k, 1., 7.}] {0.698539, 0.706226, 0.707018, 0.707098, 0.707106, 0.707107, 0.707107} ??? Who ordered this? ;-) 0.707107, 0.707107, 0.707107 ... I feel, I have seen this number elsewhere.... Omigosh! Compare! N[1/Sqrt[2]] 0.707107 Now you can see clearly that the value of the limit DEPENDS ON THE WAY OF OUT APPROACHING IT. In such a case, we say, THE LIMIT DOES NOT EXIST. ALL WE CAN SAY ABOUT IT IS THAT ITS VALUE CANNOT BE GREATER THAN YOUR 1 AND LESS THAN YOUR -1, THAT IS IT IS EQUAL TO Interval[{-1, 1}] Yesterday, I gave you the wrong number (I tried to calculate it in mind using the Stirling formula, but unfortunately had a glitch 8-( My main goal, yesterday, was to show you that your limit does not exist, and NLimit written by an outstanding calculator, Jerry Keiper, reports not what it should report (Jerry Keiper made a *PHENOMENAL* mountain of work designing and implementing the numerical components of Mathematica which we are happy to use. There is not the slightest doubt that he perfectly realized the drawback of his NLimit - but he had a Mont Blanc of tasks to do... If he were alive, he had fixed NLimit.), and it was relatively unimportant for me HOW EXACTLY YOUR LIMIT DOES NOT EXIST ;-) AS I HAVE WRITTEN YOU: The simplest way to see it might be to look at this sequence Table[Plot[Binomial[-1/2, n]*Sqrt[n*Pi], {n, 10^k, 2 10^k}], {k, 1, 4}] By the way, absence of interval concept makes the difference between the Calculus`Limit` and NumericalMath`NLimit` packages. The former uses the notion of interval (but, alas, not always can apply it successfully, as in the following example) Limit[Sin[z], z -> Infinity] Interval[{-1, 1}] << Calculus`Limit` Limit[Sin[z], z -> Infinity] Indeterminate Andrzej Kozlowski has written you on Sun, 12 May 2002 03:25:48: AK> I don't think Mathematica can do solve this problem without a lot of AK> human human help. My idea is, he is right. Thank you for your letter. Regards, Vladimir Bondarenko ........................................................................ .................. Dear Sir, Empirically the limit goes to 1 for positive even integers, and -1 for odd integers. Sqrt[n*Pi]*Binomial[-1/2, n] /. n -> {10., 100., 1000., 10000.} {0.9875829288261563, 0.9987507861261873, 0.9998750078180174, 0.999987500075049} Sqrt[n*Pi]*Binomial[-1/2, n] /. n -> {11., 101., 1001., 10001.} {-0.988704545233999, -0.9987631468267192, -0.9998751326761267, -0.9999875013126677} These appear to be the limits. Examining the graph you sent me I noticed that all the peaks were 1 and all the lows were -1 confirming the experimental data. I also received an email with a very simple algebraic proof confirming the value of the limit to be 1. James