Step 1: Understanding the Concept:
The given series can be expressed in terms of powers of 10. By observing the terms, we can identify it as an Arithmetic-Geometric Series (AGS) or use the sum formula for related power series. : Key Formula or Approach:
The general term \( T_n \) is \( 2n \cdot 10^{-(2n-1)} \).
The sum of a power series \( \sum_{n=1}^{\infty} nx^{n} = \frac{x}{(1-x)^2} \) for \( |x|<1 \). Step 2: Detailed Explanation:
Write the series in summation form:
\[ S = \frac{2}{10^1} + \frac{4}{10^3} + \frac{6}{10^5} + \frac{8}{10^7} + \dots \]
\[ S = \sum_{n=1}^{\infty} \frac{2n}{10^{2n-1}} \]
Rearrange the terms to fit the standard power series form:
\[ S = 2 \cdot 10^1 \sum_{n=1}^{\infty} \frac{n}{10^{2n}} = 20 \sum_{n=1}^{\infty} n \left( \frac{1}{100} \right)^n \]
Let \( x = \frac{1}{100} = 0.01 \).
\[ S = 20 \left[ \frac{x}{(1-x)^2} \right] \]
Substitute \( x = 0.01 \):
\[ S = 20 \left[ \frac{0.01}{(1 - 0.01)^2} \right] \]
\[ S = \frac{0.2}{(0.99)^2} = \frac{0.2}{0.9801} \]
\[ S = \frac{2000}{9801} \]. Step 3: Final Answer:
The sum of the series is \( 2000/9801 \).