Hard summation

Hi guys , i came across this summation and i couldn't solve it any ideas?

$\sum_{n=0}^\infty\frac{5^n(3^{5^{n+1}}-5\times3^{5^n}+4)}{729^{5^n}-243^{5^n}-5\times3^{5^n}+1}$

Note by Kaito Einstein
10 months, 1 week ago

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- 10 months, 1 week ago

actually it converge to $$\frac { 1 }{ 2 }$$ according to WolframAlpha

- 10 months, 1 week ago

No, it's actually slightly larger than 0.512.

- 10 months, 1 week ago

i didn't write the whole question but it says prove that it is equal to $$\frac { 1 }{ 2 }$$

- 10 months, 1 week ago

I don't know how to make it more obvious, but one can easily show by hand that $$\sum_{n=0}^1 (\text{that expression} )$$ is already larger than 0.5. Plus, (that expression) is strictly non-negative, so the infinite series is definitely larger than 0.5.

- 10 months, 1 week ago

@Chew-Seong Cheong can you help me please?!

- 10 months, 1 week ago