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I believe that the upper bound should be $L\leq (405581/10!)^{1/10} = 0.80321\cdots$, which may be compared to the actual value $L=0.7693323708\cdots$.
To apply the WZ method (or Zeilberger's algorithm) you need another parameter. You would have to find an identity with a free parameter that gives this identity as a limit or special case, and as far as I know, there's no automatic way to do this.
