<s>[INST] Prove $\sum\limits_{n=0}^\infty\frac{5^n(3^{5^{n+1}}-5\cdot3^{5^n}+4)}{(729)^{5^n}-(243)^{5^n}-5\cdot3^{5^n}+1}=\frac12$ This problem is taken from Algerian Olympiad and asks to prove that
$$\sum_{n=0}^{\infty} \dfrac{5^n(3^{5^{n+1}} -5\cdot3^{5^n} + 4)}{(729)^{5^n} - (243)^{5^n}-5\cdot3^{5^n}+1} = \frac 12.$$
Noticing that $729=3^6$ and $243 = 3^5$, I tried simplifying the general terms by setting $x=3^{5^n}$ but it seems to give no simplifications.
Thanks in advance for any advice / ideas.
 [/INST] I think the "$5 \cdot 3^{5^n}$" in the denominator is a typo and it can be proved that$$
\sum_{n = 0}^\infty \frac{5^n (3^{5^{n + 1}} - 5 \cdot 3^{5^n} + 4)}{3^{6 \cdot 5^n} - 3^{5^{n + 1}} - 3^{5^n} + 1} = \frac{1}{2}.
$$
In fact,$$
\sum_{n = 0}^\infty \frac{5^n (3^{5^{n + 1}} - 5 \cdot 3^{5^n} + 4)}{3^{6 \cdot 5^n} - 3^{5^{n + 1}} - 3^{5^n} + 1} = \sum_{n = 0}^\infty \left( \frac{5^n}{3^{5^n} - 1} - \frac{5^{n + 1}}{3^{5^{n + 1}} - 1} \right) = \frac{5^0}{3^{5^0} - 1} = \frac{1}{2}.
$$
</s>