R, 59 bytes

c=1
repeat `if`((F=F*(c=c+1)+T)>(T=T*c),F<-F-2*T/c,show(c))

Try it online!

Prints the sequence until it reaches the TIO output limit.

The numerator & denominator of the fraction so far are stored in F and T respectively. These won't error when they get too large, but R will assign them as Inf, beyond which point every integer will be (incorrectly) output, since Inf>Inf is evaluated as FALSE.

A non-overflowing version, using R+GMP to handle large integers, is 93 bytes (the R version installed on TIO seems to give an error, so here is a link to a working version on rdrr.io, with repeat exchanged for while(c<100) to force output).

R, 59 58 bytes

Edit: -1 byte thanks to emanresu A

c=1
repeat`if`((F=F*(c=c+1)+T)>(T=T*c),F<-F-2*T/c,show(c))

Try it online!

Prints the sequence until it reaches the TIO output limit.

The numerator & denominator of the fraction so far are stored in F and T respectively. These won't error when they get too large, but R will assign them as Inf, beyond which point every integer will be (incorrectly) output, since Inf>Inf is evaluated as FALSE.

A non-overflowing version, using R+GMP to handle large integers, is 93 bytes (the R version installed on TIO seems to give an error, so here is a link to a working version on rdrr.io, with repeat exchanged for while(c<100) to force output).

R, 59 bytes

c=1
repeat `if`((F=F*(c=c+1)+T)>(T=T*c),F<-F-2*T/c,show(c))

Try it online!

Prints the sequence until it reaches the TIO output limit.

The numerator & denominator of the fraction so far are stored in F and T respectively. These won't error when they get too large, but R will assign them as Inf, beyond which point every integer will be (incorrectly) output, since Inf>Inf is evaluated as FALSE.

R, 59 bytes

c=1
repeat `if`((F=F*(c=c+1)+T)>(T=T*c),F<-F-2*T/c,show(c))

Try it online!

Prints the sequence until it reaches the TIO output limit.

The numerator & denominator of the fraction so far are stored in F and T respectively. These won't error when they get too large, but R will assign them as Inf, beyond which point every integer will be (incorrectly) output, since Inf>Inf is evaluated as FALSE.

A non-overflowing version, using R+GMP to handle large integers, is 93 bytes (the R version installed on TIO seems to give an error, so here is a link to a working version on rdrr.io, with repeat exchanged for while(c<100) to force output).

R, 59 bytes

c=1
repeat `if`((F=F*(c=c+1)+T)>(T=T*c),F<-F-2*T/c,show(c))

Try it online!

R, 59 bytes

c=1
repeat `if`((F=F*(c=c+1)+T)>(T=T*c),F<-F-2*T/c,show(c))

Try it online!

Prints the sequence until it reaches the TIO output limit.

The numerator & denominator of the fraction so far are stored in F and T respectively. These won't error when they get too large, but R will assign them as Inf, beyond which point every integer will be (incorrectly) output, since Inf>Inf is evaluated as FALSE.