Summation that "feeds back into itself"

[tovisonnenberg]

Hi everyone! I need to use sigma notation to build a summation that "feeds back into itself". By that I mean that it should model a sum whose terms are
f(x) + f(f(x)) + f(f(f(x))) and so on. How would I do this?

 

[Deleted member 4993]

Hi everyone! I need to use sigma notation to build a summation that "feeds back into itself". By that I mean that it should model a sum whose terms are
f(x) + f(f(x)) + f(f(f(x))) and so on. How would I do this?

Interesting! I have not encountered this "beast" before!

Is this a class-assignment or just curiosity?

Can you provide an example of such problem?

 

[HallsofIvy]

Many texts use the notation "\(\displaystyle f^{(n)}(x)\)" to mean f composed with itself n times. Using that notation, your sum would be \(\displaystyle \sum_{i=1}^\infty f^{(i)}(x)\). (Other texts use that same notation to indicate the nth derivative of f so be sure to state your usage.)

 

[Cubist]

Many texts use the notation "\(\displaystyle f^{(n)}(x)\)" to mean f composed with itself n times. Using that notation, your sum would be \(\displaystyle \sum_{i=1}^\infty f^{(i)}(x)\). (Other texts use that same notation to indicate the nth derivative of f so be sure to state your usage.)

See the "functional powers" part of this wiki page.