Sum Of Minuscule Differences

Calculus Level 3

Let {Un}n=1n=\left \{U_n\right \}_{n=1}^{n=\infty} be a sequence of positive numbers such that n=1Un \displaystyle \sum_{n=1}^\infty U_n converges.
Is it also true that n=1(eUn1) \displaystyle \sum_{n=1}^\infty \left( e^{U_n} - 1 \right) also converges?

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