(08/01/2014, 11:36 PM)sheldonison Wrote: Of course, a graph of the Taylor series in the complex plane would be fun too, as well as the location of the zeros.

The first zero on the real line is close to -1.5, and the following recurrence gives a good approximation of the next zero, as a seed for Newton's method:

z_1 ~= -1.5

z_{n+1} ~= 2*z_n - 2^n

Interestingly, if there is a zero at z0, then there is a local minimum/maximum at 2*z0, and an inflection point at 4*z0. This follows immediately from the recursively defined derivatives:

Here are the first 50 real zeroes to double precision:

Code:

`-1.48807854559971029`

-4.88114089489665468

-13.5604085279634226

-34.7753162477509071

-84.9772901072076866

-201.002876160804470

-464.412828169898831

-1054.13189668549699

-2359.66104119548167

-5223.38043950256429

-11456.9350631199488

-24937.6038007499193

-53928.3017786735249

-115972.233276771949

-248191.706915839203

-528905.163307529388

-1.12290070098583745e6

-2.37606328140342049e6

-5.01279162501987540e6

-1.05471609094928598e7

-2.21379130993460411e7

-4.63637804012152721e7

-9.69048413062090996e7

-2.02166693910570234e8

-4.21051803671740123e8

-8.75548345417406748e8

-1.81800044561279965e9

-3.76983427215822167e9

-7.80738232697573912e9

-1.61502779267792663e10

-3.33717390504941272e10

-6.88861315522616415e10

-1.42058097312387146e11

-2.92688833903132784e11

-6.02524737808425389e11

-1.23934692808670802e12

-2.54729489811785490e12

-5.23180327153151355e12

-1.07380546760710994e13

-2.20250450743270473e13

-4.51480352072998729e13

-9.24920980897966249e13

-1.89376508958175541e14

-3.87538125916723716e14

-7.92647373212427531e14

-1.62043869049271852e15

-3.31116847010548031e15

-6.76292514833475948e15

-1.38070380849830280e16

-2.81764732177647564e16

~ Jay Daniel Fox