The infinite summations \(\sum_{n=0}^{\infty} \frac{2^{n+1}(n+1)t^n}{(n+2)!}\) and \(\sum_{n=0}^{\infty} \frac{2^{n}nt^{n-1}}{(n+1)!}\) are equal due to the properties of series manipulation. Specifically, the equality arises from the fact that \(\sum_{n=0}^\infty a_n = a_{0} + \sum_{n=0}^\infty a_{n+1}\), where \(a_0\) is zero. This relationship allows for the transformation of terms within the summations, confirming their equivalence.
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