Mathematica, 83 90 bytes

i=1;v=Tr[#*x^Range@Length@#]&;While[1<2,Echo@Limit[D[v@#/v@#2/i!,{x,i}],x->0];i++]&

Thanks to @Jenny_mathy !

Take input as two lists.

Need Alt+. to terminate the execution to see the result. Frontend may crash due to rapid output.

Limit[D[]] way is 2 bytes shorter than SeriesCofficient[]

I remember there is a built-in to replace v but I cannot find it.

Mathematica, 64 83 90 bytes

Do[Echo@Limit[D[#/#2/i!&@@Fold[x#+#2&]/@#,{x,i}],x->0],{i,∞}‌​]&

Thanks to @ngenisis and @Jenny_mathy !

Take input as two lists.

Need Alt+. to terminate the execution to see the result. Frontend may crash due to rapid output.

83 bytes version (@Jenny_mathy):

i=1;v=Tr[#*x^Range@Length@#]&;While[1<2,Echo@Limit[D[v@#/v@#2/i!,{x,i}],x->0];i++]&

Mathematica, 90 bytes

(v=Plus@@(#x^Range@Length@#)&;For[i=1,1>0,i++,Echo@(Limit[D[v[#]/v[#2]/i!,{x,i}],x->0])]&)

Take input as two lists.

Need Alt+. to terminate the execution to see the result. Frontend may crash due to rapid output.

Limit[D[]] way is 2 bytes shorter than SeriesCofficient[]

I remember there is a built-in to replace v but I cannot find it.

Mathematica, 83 90 bytes

i=1;v=Tr[#*x^Range@Length@#]&;While[1<2,Echo@Limit[D[v@#/v@#2/i!,{x,i}],x->0];i++]&

Thanks to @Jenny_mathy !

Take input as two lists.

Need Alt+. to terminate the execution to see the result. Frontend may crash due to rapid output.

Limit[D[]] way is 2 bytes shorter than SeriesCofficient[]

I remember there is a built-in to replace v but I cannot find it.