Additional data for Liviu's plots. I used Pari/GP with 1200 digits dec prec, documenting also the required number of terms after which the absolute values of the summands of the series decrease below 1e-100. There seems to be no local minimum...

$\small \begin{array} {rl|r} & & \text{# of terms}\\\ x & f(x) & \text{ required} \\\ \hline \\\ -1 & 0.438599896749 & 201 \\\ -2 & 0.247539616819 & 201 \\\ -3 & 0.162554775870 & 211 \\\ -4 & 0.117399404501 & 257 \\\ -5 & 0.0903120618145 & 304 \\\ -6 & 0.0726061182760 & 354 \\\ -7 & 0.0602796213492 & 407 \\\ -8 & 0.0512783927864 & 464 \\\ -9 & 0.0444561508357 & 525 \\\ -10 & 0.0391295513879 & 589 \\\ -11 & 0.0348689168813 & 658 \\\ -12 & 0.0313919770798 & 730 \\\ -13 & 0.0285063993737 & 808 \\\ -14 & 0.0260770215882 & 889 \\\ -15 & 0.0240063146159 & 976 \\\ -16 & 0.0222222780410 & 1067 \\\ -17 & 0.0206706877888 & 1162 \\\ -18 & 0.0193099849974 & 1263 \\\ -19 & 0.0181078191003 & 1369 \\\ -20 & 0.0170386561852 & 1479 \\\ -21 & 0.0160820905671 & 1595 \\\ -22 & 0.0152216309789 & 1715 \\\ -23 & 0.0144438135509 & 1841 \\\ -24 & 0.0137375438980 & 1972 \\\ -25 & 0.0130936024884 & 2108 \\\ -26 & 0.0125042681404 & 2250 \\\ -27 & 0.0119630281606 & 2396 \\\ -28 & 0.0114643528377 & 2548 \\\ -29 & 0.0110035182996 & 2705 \\\ -30 & 0.0105764661081 & 2867 \\\ -31 & 0.0101796910429 & 3035 \\\ -32 & 0.00981015071575 & 3208 \\\ -33 & 0.00946519223932 & 3386 \\\ -34 & 0.00914249232841 & 3569 \\\ -35 & 0.00884000806032 & 3758 \\\ -36 & 0.00855593615550 & 3953 \\\ -37 & 0.00828867911422 & 4152 \\\ -38 & 0.00803681690505 & 4357 \\\ -39 & 0.00779908317617 & 4567 \\\ -40 & 0.00757434517200 & 4783 \\\ -41 & 0.00736158670179 & 5004 \\\ -42 & 0.00715989363457 & 5231 \\\ -43 & 0.00696844149585 & 5462 \\\ -44 & 0.00678648482039 & 5700 \\\ -45 & 0.00661334797911 & 5942 \\\ -46 & 0.00644841724806 & 6190 \\\ -47 & 0.00629113392871 & 6444 \\\ -48 & 0.00614098836080 & 6703 \\\ -49 & 0.00599751469633 & 6967 \\\ -50 & 0.00586028632445 & 7236 \\\ -51 & 0.00572891185489 & 7511 \\\ -52 & 0.00560303158255 & 7792 \\\ -53 & 0.00548231436720 & 8078 \\\ -54 & 0.00536645487311 & 8369 \\\ -55 & 0.00525517112099 & 8666 \\\ -56 & 0.00514820231209 & 8968 \\\ -57 & 0.00504530688991 & 9275 \\\ -58 & 0.00494626080983 & 9588 \\\ -59 & 0.00485085599129 & 9907 \\\ -60 & 0.00475889893049 & 10230 \\\ -61 & 0.00467020945455 & 10560 \\\ -62 & 0.00458461960073 & 10894 \\\ -63 & 0.00450197260623 & 11234 \\\ -64 & 0.00442212199624 & 11580 \\\ -65 & 0.00434493075923 & 11931 \\\ -66 & 0.00427027059992 & 12287 \\\ -67 & 0.00419802126157 & 12649 \\\ -68 & 0.00412806991028 & 13016 \\\ -69 & 0.00406031057475 & 13388 \\\ -70 & 0.00399464363573 & 13766 \end{array} $

Additional data for Liviu's plots. I used Pari/GP with 1200 digits dec prec, documenting also the required number of terms after which the absolute values of the summands of the series decrease below 1e-100. There seems to be no local minimum...

$\small \begin{array}{rl|r} & & \text{# of terms}\\ x & f(x) & \text{ required} \\ \hline \\ -1 & 0.438599896749 & 201 \\ -2 & 0.247539616819 & 201 \\ -3 & 0.162554775870 & 211 \\ -4 & 0.117399404501 & 257 \\ -5 & 0.0903120618145 & 304 \\ -6 & 0.0726061182760 & 354 \\ -7 & 0.0602796213492 & 407 \\ -8 & 0.0512783927864 & 464 \\ -9 & 0.0444561508357 & 525 \\ -10 & 0.0391295513879 & 589 \\ -11 & 0.0348689168813 & 658 \\ -12 & 0.0313919770798 & 730 \\ -13 & 0.0285063993737 & 808 \\ -14 & 0.0260770215882 & 889 \\ -15 & 0.0240063146159 & 976 \\ -16 & 0.0222222780410 & 1067 \\ -17 & 0.0206706877888 & 1162 \\ -18 & 0.0193099849974 & 1263 \\ -19 & 0.0181078191003 & 1369 \\ -20 & 0.0170386561852 & 1479 \\ -21 & 0.0160820905671 & 1595 \\ -22 & 0.0152216309789 & 1715 \\ -23 & 0.0144438135509 & 1841 \\ -24 & 0.0137375438980 & 1972 \\ -25 & 0.0130936024884 & 2108 \\ -26 & 0.0125042681404 & 2250 \\ -27 & 0.0119630281606 & 2396 \\ -28 & 0.0114643528377 & 2548 \\ -29 & 0.0110035182996 & 2705 \\ -30 & 0.0105764661081 & 2867 \\ -31 & 0.0101796910429 & 3035 \\ -32 & 0.00981015071575 & 3208 \\ -33 & 0.00946519223932 & 3386 \\ -34 & 0.00914249232841 & 3569 \\ -35 & 0.00884000806032 & 3758 \\ -36 & 0.00855593615550 & 3953 \\ -37 & 0.00828867911422 & 4152 \\ -38 & 0.00803681690505 & 4357 \\ -39 & 0.00779908317617 & 4567 \\ -40 & 0.00757434517200 & 4783 \\ -41 & 0.00736158670179 & 5004 \\ -42 & 0.00715989363457 & 5231 \\ -43 & 0.00696844149585 & 5462 \\ -44 & 0.00678648482039 & 5700 \\ -45 & 0.00661334797911 & 5942 \\ -46 & 0.00644841724806 & 6190 \\ -47 & 0.00629113392871 & 6444 \\ -48 & 0.00614098836080 & 6703 \\ -49 & 0.00599751469633 & 6967 \\ -50 & 0.00586028632445 & 7236 \\ -51 & 0.00572891185489 & 7511 \\ -52 & 0.00560303158255 & 7792 \\ -53 & 0.00548231436720 & 8078 \\ -54 & 0.00536645487311 & 8369 \\ -55 & 0.00525517112099 & 8666 \\ -56 & 0.00514820231209 & 8968 \\ -57 & 0.00504530688991 & 9275 \\ -58 & 0.00494626080983 & 9588 \\ -59 & 0.00485085599129 & 9907 \\ -60 & 0.00475889893049 & 10230 \\ -61 & 0.00467020945455 & 10560 \\ -62 & 0.00458461960073 & 10894 \\ -63 & 0.00450197260623 & 11234 \\ -64 & 0.00442212199624 & 11580 \\ -65 & 0.00434493075923 & 11931 \\ -66 & 0.00427027059992 & 12287 \\ -67 & 0.00419802126157 & 12649 \\ -68 & 0.00412806991028 & 13016 \\ -69 & 0.00406031057475 & 13388 \\ -70 & 0.00399464363573 & 13766 \end{array} $