To find the sum \( C \) of the series given by

\[ C = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{9} + \frac{1}{8} + \frac{1}{27} + \ldots + \frac{1}{1024} + \frac{1}{2187}, \]

we need to identify the pattern in the series. The denominators appear to be powers of 2 and powers of 3. Let's rewrite the series by grouping terms with the same base:

\[ C = \left( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots + \frac{1}{1024} \right) + \left( \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \ldots + \frac{1}{2187} \right). \]

Each group is a geometric series. Let's analyze each group separately.

### Sum of the Series with Powers of 2

The first group is:

\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots + \frac{1}{1024}. \]

This is a geometric series with the first term \( a = \frac{1}{2} \) and common ratio \( r = \frac{1}{2} \). The number of terms can be determined by noting that \( 1024 = 2^{10} \), so there are 10 terms.

The sum \( S \) of the first \( n \) terms of a geometric series is given by:

\[ S = a \frac{1-r^n}{1-r}. \]

For our series:

\[ S = \frac{1}{2} \frac{1 - \left( \frac{1}{2} \right)^{10}}{1 - \frac{1}{2}} = \frac{1}{2} \frac{1 - \frac{1}{1024}}{\frac{1}{2}} = 1 - \frac{1}{1024}. \]

### Sum of the Series with Powers of 3

The second group is:

\[ \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \ldots + \frac{1}{2187}. \]

This is a geometric series with the first term \( a = \frac{1}{3} \) and common ratio \( r = \frac{1}{3} \). The number of terms can be determined by noting that \( 2187 = 3^7 \), so there are 7 terms.

The sum \( S \) of the first \( n \) terms of this geometric series is:

\[ S = a \frac{1-r^n}{1-r}. \]

For our series:

\[ S = \frac{1}{3} \frac{1 - \left( \frac{1}{3} \right)^7}{1 - \frac{1}{3}} = \frac{1}{3} \frac{1 - \frac{1}{2187}}{\frac{2}{3}} = \frac{1}{2} \left( 1 - \frac{1}{2187} \right). \]

### Total Sum

Now, we add the sums of the two geometric series:

\[ C = \left( 1 - \frac{1}{1024} \right) + \left( \frac{1}{2} \left( 1 - \frac{1}{2187} \right) \right). \]

Simplifying this:

\[ C = 1 - \frac{1}{1024} + \frac{1}{2} - \frac{1}{2 \cdot 2187}. \]

\[ C = 1 + \frac{1}{2} - \frac{1}{1024} - \frac{1}{4374}. \]

\[ C = \frac{3}{2} - \frac{1}{1024} - \frac{1}{4374}. \]

So, the sum of the series is:

\[ C = \frac{3}{2} - \frac{1}{1024} - \frac{1}{4374}. \]