The Easy Peasy Double Sum of Series

Calculus Level 5

The value of the following double sum of series

\[\sum_{\large m=0}^{\large\infty}\sum_{\large n=0}^{\large2014}\binom{m}{n}\left(\frac{2}{3}\right)^{\large m+1}\]

can be expressed as \(\displaystyle{\frac{2^{\large x}-2^{\large y}}{3^{\large z}}}.\) Find \(x+y+z.\)

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