Integral of a Power Series We can multiply, add and diﬀerentiate power series. Can we integrate them? Yes; as you’d expect, integration of power series is very similar to integration of polynomials. We’ll use integration to ﬁnd a power series expansion for: � x dt ln(1 + x) = (x > −1). 1 +t 0 We know that: 1 = 1 − t + t 2 − t3 + · · · 1+t So: � x ln(1 + x) (1 − t + t2 − t3 + · · ·)dt 0 � �x t2 t3 t4 = t − + − + ··· 2 3 4 0 ln(1 + x) = = x− x2 x3 x4 + − + ··· 2 3 4 Because we began with a power series whose radius of convergence was 1, the radius of convergence of the result will also be 1. This reﬂects the fact that ln(1 + x) is undeﬁned for x ≤ −1. Question: If you only use positive values of x is there still a radius of convergence? Answer: Yes. If x > 1 then the numerators x, x2 , x3 , x4 and so on are increasing exponentially. The denominators 1, 2, 3, 4 ... only grow linearly. So xn as n goes to inﬁnity, will also go to inﬁnity. If the terms of a series go to n inﬁnity then the series diverges. Euler used this kind of power series expansion to calculate natural logarithms much more eﬃciently than was previously possible.