Question related to arithmetico-geometric series

Question

The question is $$\text{Find the sum of }0.2+0.004+0.00006+0.0000008+\cdots$$ My solution: Given series can be written in the form $$S=\frac{2}{10}+\frac{4}{10^3}+\frac{6}{10^5}+\frac{8}{10^7}+\cdots(1)$$ Multiplying both sides by $10^{-2}$, we get $$\frac{S}{10^2}=\frac{2}{10^3}+\frac{4}{10^5}+\frac{6}{10^7}+\cdots(2)$$ Subtracting equation $(2)$ from equation $(1)$, we get $$S-\frac{S}{100}=\frac{2}{10}+\frac{2}{10^3}+\frac{2}{10^5}+\frac{2}{10^7}+\cdots$$ $$\implies\frac{99}{100}S=\frac{\frac{2}{10}}{1-\frac{1}{100}}=\frac{2}{10}\times\frac{100}{99}=\frac{20}{99}$$ $$\implies S=\frac{20}{99}\times\frac{100}{99}=\frac{2000}{9801}$$ But the answer given in the book is $\dfrac{2180}{9801}$. I checked my solution multiple times but can't figure out my mistake. Any help would be appreciated.

Answer 1

Alternate method, per request:

Let $\displaystyle S = \frac{2}{10} \left[1 + \frac{1}{10^2} + \frac{1}{10^4} + \cdots\right] = \frac{2}{10}\times \frac{100}{99}.$

Then the desired sum is

$$ T = S\left[1 + \frac{1}{10^2} + \frac{1}{10^4} + \cdots\right] = S \times \left(\frac{100}{99}\right) = \frac{2}{10} \times \left(\frac{100}{99}\right)^2.$$

Addendum
Actually, the underlying problem: how to evaluate
$$ g(x) = \left(1 + 2x + 3x^2 + 4x^3 + \cdots \right) ~:~ |x| < 1 \tag1$$ has 3 methods of attack. The OP and I (in effect) used closely related but distinct methods, each of intermediate difficulty.

The (3rd) easier method is to use Calculus.

Let $\displaystyle f(x) = \left(1 + x + x^2 + x^3 + \cdots\right) = \frac{1}{1-x}.$

Then $~f'(x),~$ which equals $~~g(x),~$ as shown in equation (1) above, may be equivalently expressed as

$$\frac{(-1)}{(1-x)^2} \times (-1).$$