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bobbym
2012-04-16 10:27:17

It is a general purpose method light on rigor but heavt\y on results.

anonimnystefy
2012-04-16 09:39:14

Oh,ok. I think that method can be used on another sum in the thread.

bobbym
2012-04-16 09:31:26

I am assuming they all are zero when n approaches infinity.

anonimnystefy
2012-04-16 09:30:09

But where did the rest of the infinity number of terms go?

bobbym
2012-04-16 09:25:21

So 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.

anonimnystefy
2012-04-16 09:19:43

Hi bobbym

I also used EMS,but I used the one from wiki,and it leads you to the answer in just 1 step. You have to calculate an integral and that's it.

bobbym
2012-04-16 08:15:54

This 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.