From some fiddlings with the slog-subject I came across the question, whether this is divergent or convergent:
using \( \eta = e^{1/e} = 1.44466786... \)
the sum:
\( S = \sum_{k=0}^{\infty} ( e- \eta\^ \^^k ) \)
Clearly the sequence of terms tends to zero because e is the fixpoint of iteration and in a first guess I thought that also the series converges. But the convergence of the sequence is slow and one needs a lot of terms to see a promising trend.
What I did is to look at the sequence of partial sums, each from zero to 2^n,
\( s_n = \sum_{k=0}^{2^n} ( e - \eta\^ \^^k ) \)
and that sequence {s_n} seem to increase, even slightly more than linear. at least if I look at the partial sums up to n = 12.
Here are the partial sums and the differences of order 1 to 3:
How could we prove the divergence/convergence of the series S?
Gottfried
using \( \eta = e^{1/e} = 1.44466786... \)
the sum:
\( S = \sum_{k=0}^{\infty} ( e- \eta\^ \^^k ) \)
Clearly the sequence of terms tends to zero because e is the fixpoint of iteration and in a first guess I thought that also the series converges. But the convergence of the sequence is slow and one needs a lot of terms to see a promising trend.
What I did is to look at the sequence of partial sums, each from zero to 2^n,
\( s_n = \sum_{k=0}^{2^n} ( e - \eta\^ \^^k ) \)
and that sequence {s_n} seem to increase, even slightly more than linear. at least if I look at the partial sums up to n = 12.
Here are the partial sums and the differences of order 1 to 3:
Code:
. n s_n d1_n=s_n - s_(n-1) d2_n=d1_n-d1_(n-1) d3_n
. 1 4.00875692871 4.00875692871 4.00875692871 4.00875692871
. 2 5.58578587004 1.57702894134 -2.43172798737 -6.44048491607
. 3 7.77673131247 2.19094544242 0.613916501085 3.04564448845
. 4 10.5161613469 2.73943003447 0.548484592045 -0.0654319090397
. 5 13.6651189223 3.14895757539 0.409527540923 -0.138957051122
. 6 17.0811305635 3.41601164122 0.267054065824 -0.142473475099
. 7 20.6561843799 3.57505381638 0.159042175165 -0.108011890659
. 8 24.3207908875 3.66460650761 0.0895526912324 -0.0694894839327
. 9 28.0341817806 3.71339089304 0.0487843854235 -0.0407683058090
. 10 31.7736430757 3.73946129511 0.0260704020760 -0.0227139833474
. 11 35.5268806871 3.75323761140 0.0137763162852 -0.0122940857909
. 12 39.2873487126 3.76046802550 0.00723041410300 -0.00654590218217How could we prove the divergence/convergence of the series S?
Gottfried
Gottfried Helms, Kassel