Define the sequence by and Evaluate

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- Dec 7th 2009, 09:25 PMNonCommAlgInfinite series (7)
Define the sequence by and Evaluate

- Dec 8th 2009, 02:56 AMsimplependulum
wow !

How can people ( is that you ?) find out this this expression/series for irrational numbers ?

we have

we can see that the series converges so quickly !

for this problem ,

I use a quite complicated method , is there any method better than mine ?

Go through it !

the general term of the seqence is

where

so the product of first terms is

the series can be written in this form :

To evluate this series , we just need to create next to the series .

I got

for

I got

or an elegant-appearance

the fifth term is about , wow - Dec 8th 2009, 03:44 AMNonCommAlg
- Dec 12th 2009, 04:28 PMNonCommAlg
here's another approach: the series is a "telescoping" series because

- Dec 12th 2009, 08:25 PMsimplependulum
- Dec 12th 2009, 08:30 PMsimplependulum
I have consider another method but got stuck somewhere ...

we have

so i obtain a functional equation

and i know is a one of the solution

so i let

then

i found that

but the problem is here :

is also the solution but how do we get the exact value of ? ( ) - Dec 12th 2009, 10:44 PMNonCommAlg
here's the trick: and thus therefore got it? (Smile)

- Dec 12th 2009, 11:29 PMsimplependulum