Answer :
1/3 ( e2 +
2 cos (Ö3) / e )
=
2.423641733185364535425...
This is a special case
(for z = 2, k = 3,
an = 1/n! )
of the following problem:
For an integer k and a known series
f (z) = ån an z n ,
find the value of:
The key idea is to introduce a primitive
kth root of unity, like
w = exp (2pi / k).
1 +
w +
w2 + ... +
wk-1 = 0
wk = 1
This lets the quantity 1 +
wj + ... +
w(k-1) j be k
when j is a multiple of k
and vanish otherwise.
Equating corresponding coefficients of
aj z j , we obtain:
|
f (z) +
f (w z) +
f (w2 z) + ... +
f (wk-1 z) =
k fk (z)
|
For f (z) = e z
this gives the advertised result as
f3 (2)
in the form:
1/3
[ exp (2) +
exp (2w) +
exp (2w2 ) ]
where w
= ½ (-1 + i Ö3 )
On 2008-12-26,
Dimitrina Stavrova
wrote: [edited summary]
I am greatly impressed by the quick and accurate generalization of my question, which gave me a
deeper understanding of the related material. Thank you for creating such a great site!
|
Thanks for the kind words, Dimitrina.
å xn / 3n! in
closed form
