Find the pointwise limit of the sequence$f_n(x) = {1\over 1+x} + {2\over 1+x^2} + {4\over 1+x^4}+...+{2^n\over 1+x^{2^n}}$

Question

$f_n:(1,\infty)\to\mathbb R$ is a sequence of functions defined by

$f_n(x) = {1\over 1+x} + {2\over 1+x^2} + {4\over 1+x^4}+...+{2^n\over 1+x^{2^n}}$

What is the pointwise limit of the sequence?

Since $x$ is greater than $1$, $\lim_{n\to \infty}{2^n\over 1+x^{2^n}}=0$, so at least the sequence of function should have a limit. I want to first tackle the cases where n is finite, I tried to merge the terms in $f_n$ but do not know how to do it, or are there any better ways to make $f_n$ easier to analyze?

Answer 1

HINT

I would recommend to start by noticing that

\begin{align*} \frac{1}{1+x} = \frac{1}{1 + x} + \frac{1}{1-x} - \frac{1}{1-x} = \frac{2}{1-x^{2}} - \frac{1}{1-x} \end{align*}

Similarly, we do also have that \begin{align*} \frac{2}{1-x^{2}} + \frac{2}{1+x^{2}} = \frac{4}{1 - x^{4}} \end{align*}

and so on. So the proposed series is given by: \begin{align*} f_{n}(x) = \frac{2^{n+1}}{1-x^{2^{n+1}}} - \frac{1}{1-x} \end{align*}

where $n\geq 0$. Can you take it from here?