\( \large g(a) = |X-Y| \\ \text{X = remainder when 'a' is divided by 'a+1'.} \\ \text{Y = the quotient when X is divided by a} \\ \large f(x) = \begin{cases} 0 \text{; if } x > x^2 \\ 1 \text{; if x < 2} \\ \text{else; } f(g(x)) \end{cases}\)

The above shows definition of 2 functions

Then find the value of

\(\large \color{green}{\displaystyle \sum_{n=1}^{\infty} (f(\dfrac{1}{10^n}) + f(n) - f(-n))}+ \color{blue}{1}\)

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