Find the Power Series

Question

Find the Power Series of f(x) = ∫ 0->x  e^t-1dt/t

NOTE: Assume the following: g(0) = Lim g(t) as t--->0

 

Answer 1

Find the power series of the integrand g(t) about t=0 first. Use

 

e^t = 1+t+t²/2!+t³/3!+... =∑_n=0^∞ tⁿ/n!

e^t-1 = t+t²/2!+t³/3!+... =∑_n=1^∞ tⁿ/n!

(e^t-1)/t = 1+t/2!+t²/3!+... =∑_n=1^∞ t^n-1/n!

Now integrate this series term by term:

∫₀^x (e^t-1)/t dt= ∫₀^x (1+t/2!+t²/3!+...) dt = ∫₀^x ∑_n=1^∞ t^n-1/n! dt

= x + x²/(2*2!)+x³/(3*3!)+... = ∑_n=1^∞ xⁿ/(n*n!)

By the ratio test, the series converges everywhere.