User:Henry kong

I think I need to study LaTeX

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About Me

I am Henry Kong, a 13 years-old guy.

---OK I should ask somebody to check my grammar before I publish this section---

I have (re)discovered some series that represent some constants and proved some known result using different method

Proof

Harmonic series diverges

First, we know that:

Sandbox

From Weierstrass Form:

$\ds \frac 1 {\map \Gamma z} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \left[{\left({1 + \frac z n}\right] e^{-z / n} }\right)$

We can take the reciprocal of the both side and obtain:

$\ds \map \Gamma z = \frac {e^{-\gamma z}} z \prod_{n \mathop = 1}^\infty \frac{e^{\frac z n}}{1 + \frac z n}$

Take the derivative of both side:

\(\ds \Gamma' \left({z}\right)\)

\(=\)

\(\ds -\frac{e^{-\gamma z}(1 + \gamma z)}{z^2} \prod_{n \mathop = 1}^\infty \left[{\frac{ e^{ z/n } }{(1 + \frac z n)} }\right] + \frac{ e^{-\gamma z} } z \sum_{n \mathop = 1}^\infty \left[ \frac z {n(z+n)} \prod_{i \mathop = 1}^\infty \frac{ e^{z/i} }{(1 + \frac z i)} \right]\)

General Product Rule for Derivatives

\(\ds \)

\(=\)

\(\ds -\frac{e^{-\gamma z}(1+\gamma z)}{z^2} \map \Gamma z \frac z { e^{-\gamma z} } + \frac{ e^{-\gamma z} } z \sum_{n \mathop = 1}^\infty \left[{ \frac z {n(z+n)} \frac z { e^{-\gamma z} } \map \Gamma z}\right]\)

Simplify the product notation

\(\ds \)

\(=\)

\(\ds -\frac{1+\gamma z} z \map \Gamma z + \sum_{n \mathop = 1}^\infty \left[{ \frac{z \map \Gamma z}{n(z+n)} }\right]\)

Simplify whole expression

Divide both side by $\map \Gamma z$:

\(\ds \frac{\Gamma' \left({z}\right)} {\map \Gamma z}\)

\(=\)

\(\ds -\frac{1+\gamma z} z + \sum_{n \mathop = 1}^\infty \left[{ \frac z {n(z+n)} }\right]\)

\(\ds \)

\(=\)

\(\ds -\gamma-\frac 1 z + \sum_{n \mathop = 1}^\infty \left[{ \frac 1 n - \frac 1 {z+n} }\right]\)

\(\ds \)

\(=\)

\(\ds -\gamma+\sum_{n \mathop = 1}^\infty \left[{ \frac 1 n - \frac 1 {z+n-1} }\right]\)

Rearrage the series

$\blacksquare$