Find the power series of the integrand g(t) about t=0 first. Use
e^{t} = 1+t+t^{2}/2!+t^{3}/3!+... =∑_{n=0}^{∞} t^{n}/n!
e^{t}1 = t+t^{2}/2!+t^{3}/3!+... =∑_{n=1}^{∞} t^{n}/n!
(e^{t}1)/t = 1+t/2!+t^{2}/3!+... =∑_{n=1}^{∞} t^{n1}/n!
Now integrate this series term by term:
∫_{0}^{x} (e^{t}1)/t dt= ∫_{0}^{x} (1+t/2!+t^{2}/3!+...) dt = ∫_{0}^{x} ∑_{n=1}^{∞} t^{n1}/n! dt
= x + x^{2}/(2*2!)+x^{3}/(3*3!)+... = ∑_{n=1}^{∞} x^{n}/(n*n!)
By the ratio test, the series converges everywhere.
11/3/2013

Andre W.