Inspired by Brian Charlesworth!

Calculus Level 5

\[ \large \sum_{n=0}^\infty \dfrac1{2^n\left(n+\frac14\right)} \]

If the summation above equals to \( 2^x( \tan^{-1} (\sqrt y) + \tanh^{-1} (\sqrt z)) \), find the value of \( \dfrac{2 \times x}{y\times z} \).

Also try - A naturally perfect problem.
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