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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
6 days, 18 hours ago

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What's so interesting about this sum? It doesn't appear to converge to any "nice" number. Pi Han Goh · 5 days, 11 hours ago

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@Pi Han Goh actually it converge to \(\frac { 1 }{ 2 } \) according to WolframAlpha Kaito Einstein · 5 days, 6 hours ago

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@Kaito Einstein No, it's actually slightly larger than 0.512. Pi Han Goh · 5 days, 5 hours ago

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@Pi Han Goh i didn't write the whole question but it says prove that it is equal to \(\frac { 1 }{ 2 } \) Kaito Einstein · 5 days, 3 hours ago

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@Kaito Einstein 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. Pi Han Goh · 5 days, 3 hours ago

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@Chew-Seong Cheong can you help me please?! Kaito Einstein · 5 days, 16 hours ago

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