Fun with Fractions!

Let the answer be $S$

$S = \frac{1}{10^2} + \frac{2}{10^4} + \frac{3}{10^6} + \frac{4}{10^8} + ...$

$\frac{1}{100}S = \frac{1}{10^4} + \frac{2}{10^6} + \frac{3}{10^8} + ...$

Subtracting, we get

$\frac{99}{100}S = \frac{1}{99}$

$S = \frac{100}{9801}$

There is a little wrong , the center part should be $...979900010202...$

Let $f(x) = 1+2x+3x^2+\cdots$ , then $10^{-2}f\left(10^{-2}\right) = 0.\overline{0102030405... 979900}$

$98$ is skiped since the $100^{\mathrm{th}}$ term carries $1$

$\displaystyle \int^{x}_0 f(t)\mathrm dt = x+x^2+x^3+\cdots = \frac{x}{1-x}$

$\displaystyle f(x)=\frac{\mathrm d}{\mathrm dx}\frac{x}{1-x} = \frac{1}{(1-x)^2}$

$\displaystyle 10^{-2}f\left(10^{-2}\right) = \boxed{ \displaystyle \frac{100}{9801}}$