11/01/2010, 08:32 PM
Hi.
Does this lead anywhere, or does it just fail?
Take the partial sums of \( \exp \), a function called the "exponential sum function":
\( e_n(x) = \sum_{i=0}^{n} \frac{x^i}{i!} \).
If we take this at odd values of \( n \), there will be a real fixed point. Take the regular iteration at this real fixed point, shifted so that it equals 1 at 0, call it \( \mathrm{reg}_{\mathrm{RFP}}[e_n^x](1) \), that is, regular iteration developed at the real fixed point, iterating "1" (i.e. offset so it equals 1 at 0). Now, what does
\( \lim_{k \rightarrow \infty} \mathrm{reg}_{\mathrm{RFP}}[e_{2k+1}^x](1) \)
do?
Does this lead anywhere, or does it just fail?
Take the partial sums of \( \exp \), a function called the "exponential sum function":
\( e_n(x) = \sum_{i=0}^{n} \frac{x^i}{i!} \).
If we take this at odd values of \( n \), there will be a real fixed point. Take the regular iteration at this real fixed point, shifted so that it equals 1 at 0, call it \( \mathrm{reg}_{\mathrm{RFP}}[e_n^x](1) \), that is, regular iteration developed at the real fixed point, iterating "1" (i.e. offset so it equals 1 at 0). Now, what does
\( \lim_{k \rightarrow \infty} \mathrm{reg}_{\mathrm{RFP}}[e_{2k+1}^x](1) \)
do?