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


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 column in rational format:

               0                    .
1.00000000000                    1
0.500000000000                  1/2
0.250000000000                  1/4
0                    0
-0.125000000000                 -1/8
0                    0
0.203125000000                13/64
0                    0
-0.566406250000             -145/256
0                    0
2.27343750000              291/128
0                    0
-12.1542968750            -6223/512
0                    0
82.9446411133        1358965/16384
0                    0
-703.072265625          -359973/512
0                    0
7256.32673264    1902202515/262144
0                    0
-89745.2179527  -23526170415/262144
0                    0
1312224.19186    42998962319/32768

1

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.