The series \( 1 + \frac{1 + x}{2!} + \frac{1 + x + x^2}{3!} + \frac{1 + x + x^2 + x^3}{4!} + \dots \) is equal to
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The series \( 1 + \frac{x}{2!} + \frac{x^2}{3!} + \dots \) corresponds to the Taylor series expansion of \( e^x \) minus \( e \), and dividing by \( x - 1 \) gives the final result.