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Re: "Alternating" function

Module[{i = 1},
  Nest[x + ((-1)^i++)*f[#] &, x + f[x], 4]

This does what you desire to do. Note the increment of i by one in
every Nest-iteration, and so an alternative sign before f. For a shift
in signs by one, change the starting value of i in Module by one.

Greetings from Slovenia,
Borut Levart

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