What is the value of \( 0.1+0.11+0.111+\cdots+0.1111111111 ? \)

To find the value of the series \( 0.1 + 0.11 + 0.111 + \ldots + 0.1111111111 \), we first observe the pattern in the terms of the series. We can express each term in a more manageable format. 1. The first term \( 0.1 \) can be written as: \[ 0.1 = \frac{1}{10} \] 2. The second term \( 0.11 \) can be expressed as: \[ 0.11 = \frac{11}{100} = \frac{1}{10} + \frac{1}{100} \] 3. The third term \( 0.111 \) can be expressed as: \[ 0.111 = \frac{111}{1000} = \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} \] Continuing this pattern, we can see that the \( n \)-th term \( 0.\underbrace{11\ldots1}_{n \text{ ones}} \) can be expressed as: \[ 0.\underbrace{11\ldots1}_{n \text{ ones}} = \frac{1}{10} + \frac{1}{100} + \ldots + \frac{1}{10^n} \] Therefore, we can write the \( n \)-th term as: \[ 0.\underbrace{11\ldots1}_{n \text{ ones}} = \sum_{k=1}^{n} \frac{1}{10^k} \] Now we can find the value of each term using the formula for the sum of a finite geometric series: \[ \sum_{k=1}^{n} r^k = r \left( \frac{1 - r^n}{1 - r} \right) \] For our series, \( r = \frac{1}{10} \): \[ \sum_{k=1}^{n} \frac{1}{10^k} = \frac{\frac{1}{10} \left( 1 - \left( \frac{1}{10} \right)^n \right)}{1 - \frac{1}{10}} = \frac{\frac{1}{10} \left( 1 - \frac{1}{10^n} \right)}{\frac{9}{10}} = \frac{1}{9} \left( 1 - \frac{1}{10^n} \right) \] Now, we need to sum these terms from \( n=1 \) to \( n=10 \): \[ S = 0.1 + 0.11 + 0.111 + \ldots + 0.1111111111 \] \[ S = \sum_{n=1}^{10} \sum_{k=1}^{n} \frac{1}{10^k} \] The outer sum counts how many times each \( \frac{1}{10^k} \) appears as \( k \) varies from \( 1 \) to \( n \). Specifically, \( \frac{1}{10^k} \) appears \( 11-k \) times, resulting in: \[ S = \sum_{k=1}^{10} (11-k) \cdot \frac{1}{10^k} \] Calculating each term: - For \( k=1 \), \( \frac{10}{10^1} = 1 \) - For \( k=2 \), \( \frac{9}{10^2} = 0.09 \) - For \( k=3 \), \( \frac{8}{10^3} = 0.008 \) - For \( k=4 \), \( \frac{7}{10^4} = 0.0007 \) - For \( k=5 \), \( \frac{6}{10^5} = 0.00006 \) - For \( k=6 \), \( \frac{5}{10^6} = 0.000005 \) - For \( k=7 \), \( \frac{4}{10^7} = 0.0000004 \) - For \( k=8 \), \( \frac{3}{10^8} = 0.00000003 \) - For \( k=9 \), \( \frac{2}{10^9} = 0.000000002 \) - For \( k=10 \), \( \frac{1}{10^{10}} = 0.000000