Question

We have `f:RR->RR,f(x)=e^(x-1),`and the string `(X_n)_(n>=1)`with `{(x_1=2),(x_(n+1)=f(x_n)):}`.How to demonstrate that (`X_n`) is strictly monotone?

Answer 1

See below.

Explanation:

From

https://socratic.org/questions/we-have-f-rr-rr-f-x-e-x-1-how-to-demonstrate-that-f-x-x-for-any-xinrr-1

we know that for `x_k ne 1`

`x_k < e^(x_k -1)`

so calling `x_(k+1)=e^(x_k -1)` we will have a monotonic strictly increasing sequence

`x_1 < x_2 < cdots < x_k < x_(k+1) < cdots`