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Find the Power Series

Find the Power Series of f(x) = ∫ 0->x  et-1dt/t
NOTE: Assume the following: g(0) = Lim g(t) as t--->0
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1 Answer

Find the power series of the integrand g(t) about t=0 first. Use
et = 1+t+t2/2!+t3/3!+... =∑n=0 tn/n!
et-1 = t+t2/2!+t3/3!+... =∑n=1 tn/n!
(et-1)/t = 1+t/2!+t2/3!+... =∑n=1 tn-1/n!
Now integrate this series term by term:
0x (et-1)/t dt= ∫0x (1+t/2!+t2/3!+...) dt = ∫0xn=1 tn-1/n! dt
= x + x2/(2*2!)+x3/(3*3!)+... = ∑n=1 xn/(n*n!)
By the ratio test, the series converges everywhere.