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You are not logged in. #1 20120416 08:15:54
A tough limitThis problem came up in another thread. Let's see if it can be done using a CAS and some experimental techniques. First we attempt to approximate the sum as accurately as we can: with confidence in all the digits. Next a PSLQ was done to try to identify the number in terms of simple constants. This turned out to be fruitless. So the Euler Mclaurin formula was used next. Basically the EMS is a formula that relates sums, integrals and derivatives together. It has many forms but the one we are interested in looks like this. Often this form is of use for tough sums because it is often easier to integrate and differentiate. Plugging the above sum into that equation and asking mathematica to evaluate it produces a big mess. With some work you can get this out of it. We can take the limit as n approaches infinity term by term and we are left with so and we are done. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #2 20120416 09:19:43
Re: A tough limitHi bobbym The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #3 20120416 09:25:21
Re: A tough limitSo does mine, all the remaining are just to show that they are all zero. Only the integral is is used here. Most of the time that will not be the case. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #4 20120416 09:30:09
Re: A tough limitBut where did the rest of the infinity number of terms go? The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #5 20120416 09:31:26
Re: A tough limitI am assuming they all are zero when n approaches infinity. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #6 20120416 09:39:14
Re: A tough limitOh,ok. I think that method can be used on another sum in the thread. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #7 20120416 10:27:17
Re: A tough limitIt is a general purpose method light on rigor but heavt\y on results. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. 