Note: this should go as an answer to the comment of J.C.Ottern but I want to show data and thus need the entry for answers to the original questions.
Hi J.C. - For the following I took each second coefficient. To adapt the signs I multiply by powers of $ î =\sqrt{-1} $ to get coefficients say $ d_k $. Then I show the quotients of subsequent $d_k$: $q_k = d_k/d_{k-1} = - c_{2k}/c_{2k-2}$
q_k
-1.00000000000
-0.250000000000
0.500000000000
1.62500000000
2.78846153846
4.01379310345
5.34621993127
...
< some 50 coefficients ignored >
...
350.224481320
362.230770820
374.439712641
386.851306244
399.465551130 // at k=62
412.282446834 // at k=63
To have convergence-radius >0 that ratios must converge to a constant value. But even the differences of the ratios increase:
-1.00000000000
0.750000000000
0.750000000000
1.12500000000
1.16346153846
1.22533156499
...
11.3983288747
11.6009830751
11.8036366001
12.0062894999
12.2089418203
12.4115936031
12.6142448861 // q_62-q_61
12.8168957040 // q_63-q_62
...
If that behaviour continues the generated powerseries must have convergence-radius zero.
[update] Here are the first 24 terms of the powerseries for $f(x)$ as I got them. left column in float, right middle column in most-cancelled rational format,right column normalized rational format (numerators can be found in OEIS):
0 0 .
1.00000000000 1 1/2
0.500000000000 1/2 1/4
0.250000000000 1/4 1/8
0 0 .
-0.125000000000 -1/8 -2/32
0 0 .
0.203125000000 13/64 13/128
0 0 .
-0.566406250000 -145/256 -145/512
0 0 .
2.27343750000 291/128 2328/2048
0 0 .
-12.1542968750 -6223/512 -49784/8192
0 0 .
82.9446411133 1358965/16384 1358965/32768
0 0 .
-703.072265625 -359973/512 -46076544/131072
0 0 .
7256.32673264 1902202515/262144 1902202515/524288
0 0 .
-89745.2179527 -23526170415/262144 -94104681660/2097152
0 0 .
1312224.19186 42998962319/32768 5503867176832/8388608
