Troubleshooting Newton-Raphson Iteration for Log-Likelihood Problem

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Homework Statement



Hi guys!
I have solved a log-likelihood problem and its Newton-Raphson Iteration. My step-by-step solution is attached.

The only problem I have is when I plug in numbers, the final number for the first order Newton-Raphson is different from my professor's.

I guess you don't need any numbers. If you can can detect a mistake in my formula derivation, I will really appreciate it.
 

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Just to clarify, the t index variable of the summations is distinct from \tau, right? What depends on t? What happened to the term

\sum_{t=1}^n \log x!

?
 
Yep! It's different.
I am taking derivative with respect to that t-look-alike.
So I guess you treat x as a number.
But I cannot guarantee it.
Correct me if I am wrong please.

vela said:
Just to clarify, the t index variable of the summations is distinct from \tau, right? What depends on t? What happened to the term

\sum_{t=1}^n \log x!

?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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