Taylor Series for ln(x) at a = 1

Calculus 2 · Axiom Academy

EXAMPLE Taylor Series for at a = 1 Discover the pattern in the derivatives, construct the series, and pin down its interval of convergence. Find the Taylor series for centered at a = 1 , then determine its interval of convergence. Nice work — you constructed the Taylor series for from the ground up. Here's what carried the argument: Pattern recognition: the derivatives of follow , with alternating signs. Series construction: the Taylor series is , i.e. Convergence analysis: the ratio test gives radius R = 1 , so the series converges for . Interval of convergence: after checking endpoints, the series converges for — including x = 2 but excluding x = 0 . Why x = 0 fails: there the terms reduce to , the harmonic series, which diverges (and is undefined anyway). This series is handy for logarithms of numbers near 1 : for example .

This is the written version of the interactive lesson above. See the full Calculus 2 course.