The given expression seems to represent a series where each term is the reciprocal of a certain positive integer. Let's sum the series:
\[2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \ldots\]
This is the infinite series where each term is the reciprocal of successive positive integers with denominators being powers of 2 or 3.
\[2 = \frac{2}{1}\]
So, the series starts with 2. Then,
\[\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots\]
is a geometric series with the first term \(\frac{1}{2}\) and the common ratio \(\frac{1}{2}\). The sum of this series is given by:
\[S = \frac{a}{1 - r}\]
where \(a\) is the first term and \(r\) is the common ratio. So,
\[S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1\]
Similarly,
\[\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots\]
is a geometric series with the first term \(\frac{1}{3}\) and the common ratio \(\frac{1}{3}\). The sum of this series is:
\[S = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2}\]
So, the sum of the series is:
\[2 + 1 + \frac{1}{2} = 3 + \frac{1}{2} = \frac{7}{2}\]
Therefore, the sum of the series is \(\frac{7}{2}\) or 3.5 .
