Why does this sum $\sum\limits_{n=1}^{\infty}{\frac{2^{2^{n-1}}}{2^{2^n}-1}}$ converge to 1?

Question

A friend of mine gave me this quiz: Where does this sum converge to? $$\sum\limits_{n=1}^{\infty}{\frac{2^{2^{n-1}}}{2^{2^n}-1}}$$ $a_1=\frac{2}{3} , a_2=\frac{4}{15}, a_3=\frac{16}{255},...$and $S_1=\frac{2}{3}, S_2=\frac{14}{15} , S_3=\frac{254}{255}, ...$ So I thought that this sum will converge to 1.

Let $$2^{2^{n-1}}=A_n, a_n=\frac{A_n}{A_n^2-1}=\frac{1}{2}\left(\frac{1}{A_n+1}+\frac{1}{A_n-1}\right)$$ Since $A_n = A_{n-1}^2$, I thought that this could lead to finding the partial sum of the series. I could prove that $$S_n=1-\frac{1}{A_n^2-1}$$by mathematical induction, but I want to know whether my method can be useful for finding $S_n$ from scratch.

So my question is : Is there a way to find $S_n$ from scratch??

Also, I found that $$a_n=\frac{\frac{1}{A_n}}{1-\frac{1}{A_n^2}}=\sum_{n=1}^{\infty}{\frac{1}{A_n^{2n-1}}}=\frac{1}{2^{1\cdot2^{n-1}}}+\frac{1}{2^{3\cdot 2^{n-1}}}+\frac{1}{2^{5\cdot 2^{n-1}}}+....\\ \therefore \sum_{n=1}^\infty {a_n}=\frac{1}{2^{1\cdot 2^{0}}}+\frac{1}{2^{3\cdot 2^{0}}}+\frac{1}{2^{5\cdot 2^{0}}}+...\\ +\frac{1}{2^{1\cdot 2^{1}}}+\frac{1}{2^{3\cdot 2^{1}}}+\frac{1}{2^{5\cdot 2^{1}}}+...\\ +\frac{1}{2^{1\cdot 2^{2}}}+\frac{1}{2^{3\cdot 2^{2}}}+\frac{1}{2^{5\cdot 2^{2}}}+... \\ =\sum_{n=1}^{\infty}{2^{-n}}=1$$

Is this a valid way? I used the Riemann series theorem to rearrange them.

Answer 1

Here's another manipulation you could do that doesn't require any rearranging, just the simple fact that two convergent series can be added term-wise (which is a commutativity of addition plus an addition of convergent sequences statement). Let's write $$s_n = \frac{2^{2^{n-1}}}{2^{2^n}-1};$$ $$a_n = \frac{1}{2^{2^{n-1}}-1};$$ $$b_n = \frac{1}{2^{2^{n-1}}+1};$$ $S=\sum_{n=1}^\infty s_n$; $A = \sum_{n=1}^\infty a_n$; and $B = \sum_{n=1}^\infty b_n$. It's easy to see that $A,B < \infty$ by comparison to some simple (e.g., geometric) series. Since $s_n = a_n/2 + b_n/2$, we have that $S=A/2+B/2<\infty$ and the following hold: $$a_n = a_{n-1}/2 - b_{n-1}/2, \;\;\;\mbox{so}$$ $$A = a_{0}/2-b_0/2 + A/2 - B/2 = 1+A/2-B/2, \;\;\;\mbox{so}$$ $$S = A/2 + B/2 = (1+A/2-B/2)/2+B/2 = 1/2+(A/2+B/2)/2 = 1/2 + S/2,$$ which of course implies $S=1$.