Find the sum of the series 2

Calculus Level 4

1+1+71!+1+7+722!+1+7+72+733!+\large 1+\dfrac{1+7}{1!} + \dfrac{1+7+7^2}{2!} + \dfrac{1+7+7^2+7^3}{3!} +\cdots

If the above sum can be represented in the form of aeaeb\dfrac{ae^a-e}{b}, then find the value of a+ba+b.


Clarification: e2.71828e \approx 2.71828 denotes the Euler's number.


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