Hard summation

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

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

#Calculus

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

Abdeslem Smahi - 3 years, 10 months ago

What's so interesting about this sum? It doesn't appear to converge to any "nice" number.

Pi Han Goh - 3 years, 10 months ago

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

Abdeslem Smahi - 3 years, 10 months ago

No, it's actually slightly larger than 0.512.

Pi Han Goh - 3 years, 10 months ago

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

Abdeslem Smahi - 3 years, 10 months ago

@Abdeslem Smahi – 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 - 3 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$