What kind of series? The Taylor series for $W_0(x)$ at $x=0$ has radius of convergence $1/e$, so it cannot represent anything when $x>1/e$.

There is a transseries you could use, like this
$$
W_0(x) = \log(x) - \log(\log(x)) + \frac{\log(\log(x))}{\log(x)} + \cdots
$$
as $x \to \infty$.

(comput in Maple: http://www.mapleprimes.com/questions/120720-Asymptotics-Of-Lambert-W )

**added**: More terms
$$
\ln \left( x \right) -\ln \left( \ln \left( x \right) \right) +{
\frac {\ln \left( \ln \left( x \right) \right) }{\ln \left( x
\right) }}+{\frac {-\ln \left( \ln \left( x \right) \right) +1/2\,
\left( \ln \left( \ln \left( x \right) \right) \right) ^{2}}{
\left( \ln \left( x \right) \right) ^{2}}}+{\frac {\ln \left( \ln
\left( x \right) \right) -3/2\, \left( \ln \left( \ln \left( x
\right) \right) \right) ^{2}+1/3\, \left( \ln \left( \ln \left( x
\right) \right) \right) ^{3}}{ \left( \ln \left( x \right)
\right) ^{3}}}+ \left( -\ln \left( \ln \left( x \right) \right) +3
\, \left( \ln \left( \ln \left( x \right) \right) \right) ^{2}-{
\frac {11}{6}}\, \left( \ln \left( \ln \left( x \right) \right)
\right) ^{3}+1/4\, \left( \ln \left( \ln \left( x \right) \right)
\right) ^{4} \right) \left( \ln \left( x \right) \right) ^{-4}+
\left( \ln \left( \ln \left( x \right) \right) -5\, \left( \ln
\left( \ln \left( x \right) \right) \right) ^{2}+{\frac {35}{6}}\,
\left( \ln \left( \ln \left( x \right) \right) \right) ^{3}-{
\frac {25}{12}}\, \left( \ln \left( \ln \left( x \right) \right)
\right) ^{4}+1/5\, \left( \ln \left( \ln \left( x \right) \right)
\right) ^{5} \right) \left( \ln \left( x \right) \right) ^{-5}+O
\left( {\frac { \left( \ln \left( \ln \left( x \right) \right)
\right) ^{6}}{ \left( \ln \left( x \right) \right) ^{6}}} \right)
$$

I put $x=11$ in there and got $1.80705$, but according to Maple, the true value is $W(11) = 1.80650\dots$.