Hopefully this problem will get some more good response like the one from last week. It seems like it should be a little easier, but after working at it for a few minutes, I have yet to see a possible method of solving it.
Find the infinite sum of the series:
1/1 + 1/1 + 1/2 + 1/3 + 1/5 + 1/8 + 1/13 + ..... ************************************************* 08-04-03, 04:13 PM aminator2002 As the denominator approaches infinity, the value of the series will approach 3.36
Since the number will continue to become smaller its affect on the series will become less and less.
It's very easy to put this series into a spreadsheet to see this effect. ******** 08-05-03, 08:03 AM DrGerard I'd be surprised if you could express this one in terms of known constants... The numerical value of the sum is about:
[This message was edited by methos5000 on 08-06-03 at 03:35 PM.] ******** 08-06-03, 12:39 PM FlyingHellfish Supposedly, it does have an exact answer. Unfortuately, I can't figure out a good way to begin to find it. ******** 08-06-03, 01:19 PM aminator2002 It isn't possible that this would have a sum that is a constant... the values are decreasingly increasing.
There is no repeating value to it since it is simple addition...
I just read a whole series of pages about Fibonacci series and I think the best you'll get is an approximation with an expression...
starting at the 72nd term the sum is 3.359885666243180000000000000000 but that is just rounded; it is still changing each term.
Who told you that there was a constant value? ******** 08-06-03, 02:49 PM FlyingHellfish
quote:Originally posted by aminator2002: It isn't possible that this would have a sum that is a constant... the values are decreasingly increasing.
That isn't a sufficient condition to prove that the sum isn't a constant. Take for instance the infinite sum of the series 1 + 1/2 + 1/4 + 1/8.... The sum is getting larger, only a slower rate, as you go along, but the sum of this series is a constant, specifically 2.
quote:I just read a whole series of pages about Fibonacci series and I think the best you'll get is an approximation with an expression...
I'm starting to think you're right, but....
quote:Who told you that there was a constant value?
A friend of mine who sent it to me. He didn't give a specific source, but he did say that it was a problem from a mathematical contest where calculators were not allowed. Maybe he read the problem wrong [Shrugs shoulders]. ******** 08-06-03, 03:20 PM maiku What are you guys saying?
Of course there is a constant value to the sum of the series, if it converges (which it does, since its terms are monotonically decreasing and the sum of all of the remaining terms left after we stop with a given one is less than the last term examined. What's the problem?)
The problem is to express this sum in terms of known constants, as DR. G has said. We can get arbitrarily close to the real sum, but can we write it down? In a sense, we can. I can call the number FFH and express this exactly as equal to the sum of the series he mentions. So? Where's the question?
We can write down exactly various infinite series for numbers like p, too, but we still can't write these down in decimal notation. What are you asking for here, exactly?
(And could the moderator of this forum please edit the post that is causing the margin problem?) Never mind, seems to have righted itself. ******** 08-06-03, 03:55 PM maiku Didn't right itself by itself, I see, but instead by your prompt and reliable monitoring. Good going, methos!
Now, I still don't know where aminator gets the idea that the sum of the series can't be a constant because it keeps changing. As you've pointed out, FH, a convergent series still may have a definite sum (or limit, at least), even though it keeps on getting a little bigger as we go on adding to it, and even if we can't write the limiting result of all of the additions down as a decimal or in terms of previously known constants. ******** 08-06-03, 05:05 PM aminator2002 okay maiku... it is the idea that a limit is a constant that was giving me the problem. ******** 08-06-03, 05:52 PM FlyingHellfish You're right, Maiku. I was taking liberties with the term "constant" that made it confusing. I should have said that the infinite sum of the series has an exact value (once again, supposedly), not an approximation. For anyone who's confused about the difference, an exact value is something like Ö2 or tan 60° while an approximation is the value your calculator would give for those exact values, like 1.41421356.... or 1.73205080... ******** 08-06-03, 10:07 PM DrGerard Well, FH the term "exact value" doesn't accurately reflect what you mean either... The value of this sum is just as "exact" as p or Ö2. Just consider it a rare jewel when it so happens that you can express the sum of some infinite series in terms of other well-known quantities.
Miraculous examples include either the sum of reciprocal squares (or any other even power) or the alternating sum of the reciprocal odd cubes (or any other odd power): In either case, you obtain a rational multiple of the corresponding power of p, but you'd have no such luck, say, for the straight sum of the reciprocal cubes, which is thought to be an entirely independent constant (now called Apéry's Number, after the French mathematician who first proved its irrationality in 1977).
The sum of the reciprocals of the positive Fibonacci numbers is thought to have a similarly independent status, although such statements are so difficult to prove that no one has ever done it in any "interesting" case...
I am sorry the moderator had to split over 2 lines the value I gave earlier; I (wrongly) thought the thing was short enough to fit in a single line of this layout. As my earlier post may no longer be as clear as it once was, let me make the same point again in a format that'll fit any screen, by slicing the first 2560 decimals of the expansion in groups of 10... Here's what the sum looks like. Are YOU sure you don't recognize this number? What's the next digit? 3. 3598856662 4317755317 2011302918 9271796889 0513373196 8486495553 8153251303 1899668338 3615416216 4567900872 9704534292 8853913304 1367890171 0088367959 1351733077 1190785803 3355033250 7753187599 8504871797 7789700603 9564509215 3758927752 6567335402 4033169441 7992939346 1099262625 7964647651 8686594497 1021655898 4360881472 6932495910 7947387367 3378523326 8774997627 2775794685 3676918541 9814676687 4299876738 2096913901 2177220244 0520815109 4264934951 3745416672 7895534447 0777775847 8025963407 6907484741 5557910420 0675015203 4107053352 8512979263 5242062267 5375680557 6195566972 0848843854 4079833242 9285136807 0827522662 5797511886 4646409673 7461572387 2362955620 5361220302 4635409252 6784242243 4703631036 3201466298 0402490155 7872445617 6000319551 9879059699 4202917886 6949174808 0967465236 8265408693 8399069873 2117521669 5706385941 1814553647 3642687824 6292616665 0100098903 8048233595 1989314615 0108288726 3928876699 1714930405 3057745574 3215611672 9898561772 9731395370 7352919668 8432789802 2165047585 0280918062 9100244427 7017460241 0404177860 6919006503 7142835294 2454906015 9304435523 6839237185 0498719863 0763391594 9365961429 8835472459 0992947082 6722312017 2721663872 1241823194 2461961057 7520161527 4957788183 2442620171 3211933554 2401421132 2226700585 0779033680 5573465162 5058402299 0863875041 5257469368 2112471825 6844645556 0163162771 3455714375 1455007624 8709915618 2826892640 7729981514 7332688964 9615571449 2245865861 3241975424 9306781786 2593585541 9053397178 9150545558 6364455620 7938031405 8450581455 4507217415 5386643901 3072379329 3693324593 4080446768 6577840688 8082567405 1153703481 6565354065 6305697845 3326208049 9053683459 7514423561 1198399264 7704742776 9803101105 8815581718 4923932868 9438984433 2490569195 4917268760 9559508041 2484819283 2597163873 0802925381 9599769804 1544855121 7281602262 0331943066 6297327595 3192227665 9735532353 6048195252 1028670014 2139979788 4243553601 6795991220 4261837046 1574804015 1050250099 9175943684 6692219596 0627316577 6487679414 8052152436 3689518527 4413814062 6600534344 1160130978 5072730856 4103745364 9905644909 2564022479 2358931548 7176694924 3670359504 3022373000 3069311959 1523084241 2942958779 7826500093 7094948239 5881794956 5175795285 9542579393 8129592869 0136021009 4380436689 5230969934 1361290441 7034628264 1703783983 5745622337 0388719700 9716007655 7065423903 8874719113 5079115407 6113460540 5981579190 2305756264 6715753591 6491877701 4875545821 8502626932 4924055408 5215622504 4719698236 6623355498 6281689696 6683627789 0783224455 2306778661 1861907014 6699590190 4809618163 9899148842 7181733025 7799164004 7704791863 4507268563 9394496425 9507568329 2036957858 7382609087 7061694844 3545099148 6528604325 6563839001 6063340306 1156303539 6354098363 2188358344 7308088390+ [This message was edited by DrGerard on 08-06-03 at 10:30 PM.] ******** 08-07-03, 09:25 AM maiku Or, letting F represent the number in question, it can be written exactly as
http://home.intouchmi.com/maiku/equations/Fib.gif ******** 08-08-03, 08:07 AM FlyingHellfish My apologies...the question relayed to me wasn't the original question that was meant to be solved. ******** 08-20-03, 03:03 PM maiku My apologies, too, in advance, if I am belaboring this point unnecessarily. I thought it was important, though, to get clear about a confusion which often arises whenever irrational numbers (which by definition can't be written down exactly as terminating or repeating decimals) are involved. The numbers Ö2 and p are two such well-known beasts. These cannot be written down as decimals, but the diagonal of a unit square and the ratio of a circle to its diameter are nevertheless both perfectly definite, exact quantities. It is mistaken to think, as I have found many people do, that the value of p is merely an approximate one, based on something like the precision of our measurements of circumferences and radii. Not so. We know exactly what the value of p is, though we also know we can never write it down in decimal notation. It is just because this is such a remarkable truth about mathematics that it deserves to be stressed. ******** 08-22-03, 01:19 AM silenteuphony I think most people find rational numbers more "real" than irrational numbers because they tend to relate numbers most strongly to one-dimensional measurements, and it's much easier to mark a one-dimensional ruler in equal, rational increments than in irrational ones.
In other words, it's easy to relate the number 1.6 to a length of 1 and 6/10 units on a ruler marked in tenths, but it's a lot harder to relate pi to a mark on a ruler, even though it's just as valid a number.
However, if we expand to two dimensions, it's easy to relate irrational numbers to simple, real examples. For example, if you want to "see" pi, just find (or construct) a circle with radius 1, and look at its area. Or if you want to "see" the square root of 2, just take a 2 x 1 rectangle, reshape it into a square (without changing the area), and look at the length of the side.
Or, as maiku has already pointed out, there are equally valid examples using just one-dimensional measurements, i.e. the perimeter of a circle with a diameter of 1, or the diagonal of a square with a width of 1. However, even in these examples, it's necessary to use 2-dimensional figures, which requires a greater level of abstraction than using a simple, incrementally-marked ruler to generate rational numbers. ******** 08-22-03, 06:37 PM maiku You're absolutely right, SE. The very concept of irrational numbers had to await the development of plane geometry, and it is not something the study of arithmetic by itself would ever likely have stumbled upon. Do not forget that the Greeks called irrational numbers incommensurables, and for a good reason: they arose when you compared a measurement in one direction with another in a different one and found they never came out as multiples of each other. These kinds of numbers never arose when you only measured something in one dimension. Rational numbers suffice for linear measurement.
For that matter, though, they suffice for all practical purposes for everything else, too. All numbers representable by computers, even the latest state-of-the-art ones, are really rational numbers. You can't count irrational numbers, as Cantor showed. What I find especially fascinating is that we can know, in principle, what a certain irrational (or even transcendental) number is, without being able to count it out, in terms of any ratio of discrete lumps of things. I find in this fact a powerful argument against a philosphy of materialism in any form.This message has been edited. Last edited by: DorianGreyed,
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