Proof of absolute convergence

Question

I am independently studying calculus using MIT's publicly available materials on OCW. One of the Final Exam practice question is the following:

Suppose the series $\sum_{n=1}^{\infty}a_n$ converges absolutely. Prove $\sum_{n=1}^{\infty}(e^{a_n} - 1)$ converges absolutely.

Since the rest of the practice problems have been more or less trivial, I suspect that I am missing something obvious, but I am not sure what. A hint -- the lighter the better, please! -- would be most welcome...

Answer 1

Hint:

Use the series expansion of the exponential function: $$e^{a_n} -1 = a_n + \frac{1}{2}a_n ^2 +...$$