Sum Of Minuscule Differences

Calculus Level 3

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

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